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Managing EditorLinus Kramer [email protected]

EditorsMartin Burger mburg [email protected] Cuntz [email protected] Echterhoff [email protected] Lowe [email protected] Luck [email protected] Schindler [email protected] Schneider [email protected] Tent [email protected] Wilking [email protected]

Technical EditorWend Werner [email protected]

Das Munster Journal of Mathematics wird vom Fachbereich Mathematik undInformatik der WWU Munster herausgegeben. Erscheinungsort ist Munster.

Managing Editor:Linus KramerMath. Institut, Universitat Munster, Einsteinstr. 62, 48149 Munster, Germany.

Typesetting in LATEX.

ISSN 1867-5778 (Print)ISSN 1867-5786 (Internet)

Contents

1. The EditorsEditorial 3

2. Pere Ara, Miquel Brustenga, and Guillermo CortinasK-theory of Leavitt path algebras 5

3. Marius Dadarlat and Mikael RørdamStrongly self-absorbing C∗-algebras which contain a nontrivialprojection 35

4. Christopher DeningerMahler measures and Fuglede-Kadison determinants 45

5. Siegfried Echterhoff and Oliver PfanteEquivariant K-theory of finite dimensional real vector spaces 65

6. David E. Evans and Mathew PughOcneanu cells and Boltzmann weights for the SU(3) ADE graphs 95

7. Michael Joachim and Stephan StolzAn enrichment of KK -theory over the category of symmetricspectra 143

8. Lars KadisonSkew Hopf algebras, irreducible extensions and the Π-method 183

9. Wolfgang LuckOn the classifying space of the family of virtually cyclic subgroupsor CAT(0)-groups 201

10. Ralf Meyer and Ryszard NestC∗-algebras over topological spaces: the bootstrap class 215

11. Michael PuschniggDie universelle unbeschrankte Derivation 253

12. Marc A. RieffelDirac operators for coadjoint orbits of compact Lie groups 265

ii

13. Erling StørmerDuality of cones of positive maps 299

14. Wilhelm Winter and Joachim ZachariasCompletely positive maps of order zero 311

Munster J. of Math. 2 (2009), 3–4 Munster Journal of Mathematics

urn:nbn:de:hbz:6-10569528254 c© Munster J. of Math. 2009

Editorial

This special issue of the Munster Journal of Mathematics is dedicated toour colleague and friend, Professor Joachim Cuntz, on the occasion of his 60thbirthday. To celebrate this event we organized a workshop on Noncommuta-tive Geometry, September 24–27, 2008, at the Department of Mathematics inMunster. Many of the participants at that conference are among the contrib-utors to this special issue.

The work of Joachim Cuntz has had an enormous influence on the theory ofOperator Algebras and Noncommutative Geometry. Some of the first exam-ples of C*-algebras that a student in this field learns about are certainly thefamous Cuntz algebras and their generalizations, the Cuntz-Krieger algebras.In 1981 he introduced the notion of a purely infinite C*-algebra, which nowplays a prominent role in the classification theory of C*-algebras, as does theCuntz semigroup, an important invariant. His various pictures of Kasparov’sbivariant K-theory have deepened our understanding of this work. More re-cently, he has succeeded in extending the power of bivariant K-theories to verygeneral classes of topological algebras outside the rigid world of C*-algebras.In joint work with Quillen he obtained new descriptions of cyclic homology andgave a proof of the important excision property in bivariant cyclic cohomology.These are just some of the areas in which the work of Joachim Cuntz has hada deep and lasting influence.

With this special issue of the Munster Journal of Mathematics we wouldlike to congratulate Joachim Cuntz on his many fundamental contributions tomathematics and wish him many more productive and creative years to come!

The Editors



K-theory of Leavitt path algebras

Pere Ara, Miquel Brustenga, and Guillermo Cortinas

(Communicated by Siegfried Echterhoff)

Dedicated to Joachim Cuntz on the occasion of his 60th birthday

Abstract. Let E be a row-finite quiver and let E0 be the set of vertices of E; consider theadjacency matrix N ′

E= (nij) ∈ Z(E0×E0), nij = # arrows from i to j. Write Nt

Eand

1 for the matrices ∈ Z(E0×E0\Sink(E)) which result from N ′tE and from the identity matrix

after removing the columns corresponding to sinks. We consider the K-theory of the Leavittalgebra LR(E) = LZ(E) ⊗ R. We show that if R is either a Noetherian regular ring or astable C∗-algebra, then there is an exact sequence (n ∈ Z)

Kn(R)(E0\Sink(E))1−Nt

E // Kn(R)(E0) // Kn(LR(E)) // Kn−1(R)(E0\Sink(E)) .

We also show that for general R, the obstruction for having a sequence as above is mea-sured by twisted nil-K-groups. If we replace K-theory by homotopy algebraic K-theory, theobstructions disappear, and we get, for every ring R, a long exact sequence

KHn(R)(E0\Sink(E)) 1−NtE−→ KHn(R)(E0) → KHn(LR(E)) → KHn−1(R)(E0\Sink(E)).

We also compare, for a C∗-algebra A, the algebraic K-theory of LA(E) with the topologicalK-theory of the Cuntz-Krieger algebra C∗

A(E). We show that the map

Kn(LA (E)) → Ktopn (C∗

A (E))

is an isomorphism if A is stable and n ∈ Z, and also if A = C, n ≥ 0, E is finite with nosinks, and det(1 − Nt

E) 6= 0.

1. Introduction

Cuntz and Krieger [15] generalized the construction of the Cuntz algebrasOn of [11] by considering a class of C∗-algebras associated to finite squarematrices with entries in 0, 1. Subsequently, it was realized that the Cuntz-Krieger algebras were specific cases of a more general C∗-algebra structure,

The first and second named authors were partially supported by DGI MICIIN-FEDERMTM2008-06201-C02-01, and by the Comissionat per Universitats i Recerca de la Gener-alitat de Catalunya. The third named author was supported by CONICET and partially

supported by grants PICT 2006-00836, UBACyT X051, and MTM2007-64074.

6 Pere Ara, Miquel Brustenga, and Guillermo Cortinas

the graph C∗-algebras defined in [35] and then initially studied in depth in[22]. We refer the reader to [25] for further information on this important classof C∗-algebras. Leavitt path algebras, a natural algebraic version of Cuntz-Krieger graph C∗-algebras, were introduced and studied firstly in [1] and [4].These algebras generalize the classical Leavitt algebras of type (1, n), studiedby Leavitt in [24], in much the same way as graph C∗-algebras generalize theclassical Cuntz algebras On.

In this paper, we consider theK-theory of the Leavitt path algebra LR(E) =LZ(E)⊗R of a row-finite quiver E with coefficients in a ring R. To state ourresults, we need some notation. Let E0 be the set of vertices of E; considerthe adjacency matrix N ′E = (nij) ∈ Z(E0×E0), nij = # arrows from i to j.Write N t

E and 1 for the matrices ∈ Z(E0×E0\Sink(E)) which result from N ′tE andfrom the identity matrix after removing the columns corresponding to sinks.Our results relate the K-theory of LR(E) with the spectrum

C = hocofiber(K(R)(E0\Sink(E)) 1−NtE−→ K(R)(E0)).

In terms of homotopy groups, the fundamental property of C is that there isa long exact sequence (n ∈ Z)(1.1)

Kn(R)(E0\Sink(E))1−NtE // Kn(R)(E0) // πn(C) // Kn−1(R)(E0\Sink(E)).

For a rather general class of rings (which includes all unital ones) and all row-finite quivers E, we show (Theorem 6.3) that there is a naturally split injectivemap

(1.2) π∗(C)→ K∗(LR(E)).

The cokernel of (1.2) can be described in terms of twisted nil-K-groups (see5.10, 6.6). We show that these nil-K-groups vanish for some classes of ringsR, including the following two cases:

• R is a regular supercoherent ring (see 7.6). In particular this coversthe case where R is a Noetherian regular ring.• R is a stable C∗-algebra (see 9.12).

In particular for such R we get a long exact sequence

(1.3)

Kn(R)(E0\Sink(E)) 1−NtE−→ Kn(R)(E0) → Kn(LR(E))→ Kn−1(R)(E0\Sink(E)).

To get these results, we use a blend of algebraic techniques and techniquesadapted from the analytic setting. For a finite quiver without sinks E, anadaptation of the methods of [12] and [13] allows us to apply work of Graysonand Yao about theK-theory of twisted polynomial rings ([19], [39]), to computethe K-theory of LR(E). The general row-finite case follows then from a colimitargument. The results of Waldhausen [34] are then used to derive the vanishingof the twisted nil-K-groups in the case of a regular supercoherent coefficientring R.

Munster Journal of Mathematics Vol. 2 (2009), 5–34

Leavitt path algebras 7

We also consider Weibel’s homotopy algebraic K-theory KH∗(LR(E)). Weshow in 8.6 that for any ring R and any row-finite quiver, there is a long exactsequence

(1.4) KHn(R)(E0\Sink(E)) 1−NtE−→ KHn(R)(E0) → KHn(LR(E))

→ KHn−1(R)(E0\Sink(E)).

There is a natural comparison map K∗ → KH∗; if R is a regular supercoher-ent ring or a stable C∗-algebra, then K∗(R) → KH∗(R) and K∗(LR(E)) →KH∗(LR(E)) are isomorphisms, so the sequences agree in these cases. Wefurther compare, for a C∗-algebra A, the algebraic K-theory of LA(E) withthe topological K-theory of the Cuntz-Krieger algebra C∗A(E); we show thatthe natural map

γAn (E) : Kn(LA(E))→ Kn(C

∗A(E))→ Ktop

n (C∗A(E))

is an isomorphism in some cases, including the following two:

• A = C, E is finite with no sinks, det(1−N tE) 6= 0, and n ≥ 0 (see 9.4).

• A is stable, E is row-finite, and n ∈ Z (see 9.13).

The rest of this paper is organized as follows. In Section 2 we recall theresults of Suslin and Wodzicki on excision in K-theory and draw some conse-quences which are used further on in the article. The most general result onexcision in K-theory, due to Suslin [30], characterizes those rings A on whichK-theory satisfies excision in terms of the vanishing of Tor groups over theunitization A = A⊕ Z. Namely A satisfies excision if and only if

(1.5) TorA∗ (Z, A) = 0 (∗ ≥ 0).

We call a ring A H ′-unital if it satisfies (1.5); if A is torsion-free as an abeliangroup, this is the same as saying that R is H-unital in the sense of Wodzicki[38]. We show in Proposition 2.8 that if A is H ′-unital and φ : A → A is anautomorphism, then the same is true of both the twisted polynomial ring A[t, φ]and the twisted Laurent polynomial ring A[t, t−1, φ]. We recall that, for unitalA, the K-theory of the twisted Laurent polynomials was computed in [19] and[39]. If R is a unital ring and φ : R→ pRp is a corner isomorphism, the twistedLaurent polynomial ring is not defined, but the corresponding object is thecorner skew Laurent polynomial ring R[t+, t−, φ] of [3]. In Section 3 we use theresults of [39] and of Section 2 to compute the K-theory of R⊗A[t+, t−, φ⊗1]for (R, φ) as above, and A any nonunital algebra such that R ⊗ A is H ′-unital (Theorem 3.6). In the next section we consider the relation betweentwo possible ways of defining the incidence matrix of a finite quiver, and showthat the sequences of the form (1.1) obtained with either of them are essentiallyequivalent (Proposition 4.4). In Section 5 we use the results of the previoussections to compute the K-theory of the Leavitt algebra of a finite quiverwith no sources with coefficients in an H ′-unital ring (Theorem 5.10). Thegeneral case of row-finite quivers is the subject of Section 6. Our most generalresult is Theorem 6.3, where the existence of the split injective map (1.2) is



proved for the Leavitt algebra LA(E) of a row-finite quiver E. In the lattertheorem, A is required to be either a ring with local units, or a Z-torsion freeH ′-unital ring. In Section 7 we specialize to the case of Leavitt algebras withregular supercoherent coefficient rings. We show that the sequence (1.3) holdswhenever R is regular supercoherent (Theorem 7.6). For example this holdsif R is a field, since fields are regular supercoherent; this particular case, forfinite E, is used in [2] to compute the K-theory of the algebra QR(E) obtainedfrom LR(E) after inverting all square matrices with coefficients in the pathalgebra PR(E) which are sent to invertible matrices by the augmentation mapPR(E)→ RE0 . Section 8 is devoted to homotopy algebraicK-theory, KH . Fora unital ring R, a corner isomorphism φ : R→ pRp, and a ring A, we computethe KH-theory of R ⊗ A[t+, t−, φ ⊗ 1] (Theorem 8.4). Then we use this toestablish the sequence (1.4) for any row finite quiver E and any coefficientring A (Theorem 8.6). In the last section we compare the K-theory of theLeavitt algebra LA(E) with coefficients in a C∗-algebra A with the topologicalK-theory of the corresponding Cuntz-Krieger algebra C∗A(E). In Theorem9.1 we establish the spectrum-level version of the well-known calculation ofthe topological K-theory of the Cuntz-Krieger algebra C∗A(E) of a row-finitequiver E with coefficients in a C∗-algebra A. Theorem 9.4 shows that if E isa finite quiver without sinks and such that det(1−N t

E) 6= 0, then the naturalmap γC

n : Kn(LC(E))→ Ktopn (C∗C(E)) is an isomorphism for n ≥ 0 and the zero

map for n ≤ −1. In Theorem 9.13 we show that if B is a stable C∗-algebra,then γB

n is an isomorphism for all n ∈ Z.In various parts of this paper (e.g. in Section 3 or in the proofs of 8.4 and

9.1), we shall make use of the formalism of triangulated categories. For anintroduction to this subject, the reader may consult [23], for example.

2. H ′-unital rings and skew polynomial extensions

Let R be a ring and R = R⊕ Z its unitization. We say that R is H ′-unitalif

TorR∗ (R,Z) = 0 (∗ ≥ 0).

Note that, for any, not necessarily H ′-unital ring R,

TorR∗ (Z, R) = TorR∗+1(Z,Z) = TorR∗ (R,Z) (∗ ≥ 0).

Thus all these Tor groups vanish when R is H ′-unital; moreover, in that casewe also have

TorR∗ (R,R) = 0 (∗ ≥ 1), TorR0 (R,R) = R2 = R.

A right module M over a ring R is called H ′-unitary if TorR∗ (M,Z) = 0. Thedefinition of H ′-unitary for left modules is the obvious one.

Example 2.1. If R is H ′-unital then it is both right and left H ′-unitary asa module over itself. Let φ : R → R be an endomorphism. Consider thebimodule φR with left multiplication given by a ·x = φ(a)x and the usual rightmultiplication. As a right module, φR ∼= R, whence it is right H ′-unitary. If



moreover φ is an isomorphism, then it is also isomorphic to R as a left module,via φ, and is thus left H ′-unitary too.

Remark 2.2. The notion of H ′-unitality is a close relative of the notion ofH-unitality introduced by Wodzicki in [38]. The latter notion depends on afunctorial complex Cbar(A), the bar complex of A; we have Cbar

n (A) = A⊗n+1.The ring A is calledH-unital if for all abelian groups V , the complex Cbar(A)⊗V is acyclic. If A is flat as a Z-module, then Cbar(A) is a complex of flat Z-

modules and H∗(Cbar(A)) = TorA∗ (Z, A). Hence H ′-unitality is the same as

H-unitality for rings which are flat as Z-modules. Unital rings are both H andH ′-unital. Because Cbar commutes with filtering colimits, the class of H-unitalrings is closed under such colimits. Similarly, there is also a functorial complex

which computes TorA(Z, A) and which commutes with filtering colimits ([7,6.4.3]); hence also the class of H ′-unital rings is closed under filtering colimits.If A is H or H ′-unital then the same is true of the matrix ring MnA. In theH-unital case, this is proved in [38, 9.8]; the H ′-unital case follows from atheorem of Suslin cited below (Theorem 2.6). The class of H-unital rings isfurthermore closed under tensor products, by [32, 7.10]. Hence the class offlat H ′-unital rings is closed under tensor products, since it coincides with theclass of flat H-unital rings.

Lemma 2.3. Let A be a ring. If A is H ′-unital, then A⊗Q is H ′-unital.

Proof. Tensoring with Q over Z is an exact functor from A-modules to A⊗Q-modules which preserves free modules. Hence if L→ A is a free A-resolution,then L⊗Q→ A⊗Q is a free A⊗Q-resolution. Moreover,

Q⊗A⊗Q L⊗Q = L⊗Q/A · L⊗Q = (L/A · L)⊗Q.

Hence

TorA⊗Q∗ (Q, A⊗Q) = TorA∗ (Z, A) ⊗Q.

Thus A H ′-unital implies that 0 = TorA⊗Q∗ (Q, A ⊗ Q). But by [38, §2],

TorA⊗Q∗ (Q, A⊗Q) = H∗(C

bar(A⊗Q)). Thus A⊗Q is H-unital, and thereforeH ′-unital.

Corollary 2.4. If A and B are H ′-unital, and B is a Q-algebra, then A⊗Bis H ′-unital.

Proof. It follows from the previous lemma and from the fact (proved in [32,7.10]) that the tensor product of H-unital Q-algebras is H-unital.

Example 2.5. The basic examples of H ′-unital rings we shall be concerned withare unital rings and C∗-algebras. The fact that the latter are H ′-unital followsfrom the results of [30] and [32] (see [7, 6.5.2] and Theorem 2.6 below). If A isan H ′-unital ring and B a C∗-algebra, then A⊗B is H ′-unital, by Corollary2.4.



A ring R is said to satisfy excision in K-theory if for every embedding R ⊳ Sof R as a two-sided ideal of a unital ring S, the map K(R) = K(R : R) →K(S : R) is an equivalence. One can show (see e.g. [6, 1.3]) that if R satisfiesexcision in K-theory and R is an ideal in a nonunital ring T , then the mapK(R)→ K(T : R) is an equivalence too.

The main result about H ′-unital rings which we shall need is the following.

Theorem 2.6. ([30]) A ring R is H ′-unital if and only if it satisfies excisionin K-theory.

Using the theorem above we get the following Morita invariance result forH ′-unital rings.

Lemma 2.7. Let R be a unital ring, e ∈ R an idempotent. Assume e is full,that is, assume ReR = R. Further let A be a ring such that both R ⊗ A andeRe⊗A are H ′-unital. Then the inclusion map eRe⊗A→ R⊗A induces anequivalence K(eRe⊗A)→ K(R⊗A).

Proof. Put S = R⊗A, and consider the idempotent f = e⊗1 ∈ S. One checksthat f is a full idempotent, so that K(fSf) → K(S) is an equivalence. Nowapply excision.

Proposition 2.8. Let R be a ring and φ : R→ R an automorphism. AssumeR is H ′-unital. Then R[t, φ] and R[t, t−1, φ] are H ′-unital rings.

Proof. If P is a projective resolution of R as a right R-module, then P ⊗RR[t, φ] is a complex of right R[t, φ]- projective modules. Moreover, we have anisomorphism of R-bimodules

R[t, φ] = R⊕∞⊕

n=1

Rtn ∼= R⊕∞⊕

n=1

Rφn .

Thus, because R is assumed H ′-unital,

H∗(P ⊗R R) = TorR∗ (R,R) =

0 ∗ ≥ 1R ∗ = 0.

Here we have used only the left module structure of R; the identities above arecompatible with any right module structure, and in particular with both theusual one and that induced by φn. It follows that

Q = P ⊗R R[t, φ]

is a projective resolution of R[t, φ] as a right R[t, φ]-module. Since R→ R[t, φ]is compatible with augmentations, we have

Q⊗R[t,φ]

Z = P ⊗R Z.

Hence R[t, φ] is H ′-unital. Next we consider the case of the skew Laurentpolynomials. We have a bimodule isomorphism

˜R[t, t−1, φ] = R ⊕∞⊕

n=1

(Rtn ⊕ t−nR) ∼= R⊕∞⊕

n=1

(Rφn ⊕ φnR).



Thus since φnR is left H ′-unitary, the same argument as above shows thatR[t, t−1, φ] is H ′-unital.

3. K-theory of twisted Laurent polynomials

Let X , N+, N− and Z be objects in a triangulated category T . Let φ :X → X and j± : X ⊕ N± → Z be maps in T . Let i± : X → X ⊕N± be theinclusion maps. Define a map

ψ =

[i+ i+

i− i− φ

]: X ⊕X → (X ⊕N+)⊕ (X ⊕N−).

Lemma 3.1. Assume

X ⊕X ψ // (X ⊕N+)⊕ (X ⊕N−)[j+,j−]// Z

∂ // ΣX ⊕ ΣX

is an exact triangle in T . Then

(3.2) X[0,1−φ,0] // N+ ⊕X ⊕N−

[j+|N+,−j−|X ,j

−|N−]// Z

∂′

// ΣX

is an exact triangle in T , for suitable ∂′. In particular,

Z ∼= N+ ⊕N− ⊕ cone(1− φ : X → X).

Proof. Note that

ψ =

1 10 01 φ0 0

.

Consider the maps

ψ1 =

1 0 0 00 1 0 01 0 −1 00 0 0 1

, ψ2 =

1 00 00 1− φ0 0

, and ψ3 =

[1 10 1

].

We remark that ψ = ψ1ψ2ψ3. There is an exact triangle

X ⊕X ψ2 // (X ⊕N+)⊕ (X ⊕N−)j // Z

∂′′

// ΣX ⊕ ΣX.

Here j = [j+, j−]ψ1 and ∂′′ = ψ3∂. The result follows.

Let φ : X → X be a map of spectra. We write φ−1X for the colimit of thefollowing direct system

Xφ // X

φ // Xφ // X

φ // . . .

Lemma 3.3. Let X and φ be as above, and consider the map φ : φ−1X →φ−1X induced by φ. Then

hocofiber(1 − φ : X → X) ∼= hocofiber(1 − φ : φ−1X → φ−1X)



Proof. Write φ : φ−1X → φ−1X for the induced map; we have a homotopycommutative diagram

(3.4) φ−1X1−φ // φ−1X // hocofiber(1− φ)

X1−φ //

OO

X

OO

// hocofiber(1− φ).

f

OO

Both the top and bottom rows are fibration sequences. We have to show thatthe map of stable homotopy groups fn : πn hocofiber(1−φ)→ πn hocofiber(1−φ) induced by f is an isomorphism. Denote by φn the endomorphism of πn(X)induced by φ. Note that φn induces a Z[t]-action on πnX , and that

πn(φ−1X) = Z[t, t−1]⊗Z[t] πnX =: φ−1n πnX.

It follows that the long exact sequence of homotopy groups associated to thetop fibration of (3.4) is the result of applying the functor Z[t, t−1]⊗Z[t] to thatof the bottom. In particular the left and right vertical maps in the diagrambelow are isomorphisms

0→ coker (1− φn) // πn(hocofiber(1− φ)) // ker (1− φn+1)→ 0

0→ coker (1− φn)

OO

// πn(hocofiber(1− φ))

fn

OO

// ker (1− φn+1)→ 0.

OO

It follows that f is an equivalence, as wanted.

It will be useful to introduce the following notation.

Notation 3.4.1. Let A be a unital ring and let φ : A→ A be an automorphism.Define NK(A, φ)+ = hocofiber(K(A) → K(A[t, φ])) and NK(A, φ)− =hocofiber(K(A)→ K(A[t, φ−1])). We have

K(A[t, φ]) = K(A)⊕NK(A, φ)+, K(A[t, φ−1]) = K(A)⊕NK(A, φ)−.

Now let A be an arbitrary ring and let φ : A→ A be an endomorphism. WriteB = φ−1A for the colimit of the inductive system

Aφ // A

φ // Aφ // . . . .

Then φ induces an automorphism φ : B → B and we can extend it to theunitization B. Put

NK(A, φ)+ := NK(B, φ)+, NK(A, φ)− := NK(B, φ)−,

so that K(B[t, φ]) = K(B) ⊕NK(A, φ)+ and similarly for K(B[t, φ−1]). Ob-serve that this definition of NK(A, φ)± agrees with the above when A is unital

and φ is an automorphism. Moreover we have NK(A, φ)± = NK(B, φ)±.

Lemma 3.5. Let A be H ′-unital, φ : A → A an endomorphism, and B =φ−1A. Then K(B[t, φ±1]) ∼= φ−1K(A)⊕NK(A, φ)±.



Proof. We have an exact sequence

0→ B[t, φ±1]→ ˜B[t, φ±1]→ Z[t]→ 0.

By Proposition 2.8, the ring B[t, φ±1] is H ′-unital. Hence K(B[t, φ±1]) =K(B) ⊕ NK(B, φ)±, by excision. Next, the fact that K-theory preserves fil-tering colimits (see [36, IV.6] for the unital case; the nonunital case followsfrom the unital case by using that unitization preserves colimits—becauseit has a right adjoint—and that K(A) = K(A : A)) implies that K(B) ∼=φ−1K(A).

We shall make use of the construction of the corner skew Laurent polynomialring S[t+, t−, φ], for a corner-isomorphism φ : S → pSp; see [3].

Theorem 3.6. Let R be a unital ring and let A be a ring. Let φ : R → pRpbe a corner-isomorphism. Assume that R ⊗ A is H ′-unital. Then there is ahomotopy fibration of nonconnective spectra

K(R⊗A)1−φ⊗1 // K(R⊗A)⊕NK(R⊗A, φ⊗ 1)+ ⊕NK(R⊗A, φ⊗ 1)−

// K((R⊗A)[t+, t−, φ⊗ 1]).

In other words,

K((R⊗A)[t+, t−, φ⊗ 1]) = NK(R⊗A, φ⊗ 1)+ ⊕NK(R⊗A, φ⊗ 1)−

⊕ hocofiber(K(R⊗A)1−φ⊗1−→ K(R⊗A)).

Proof. Step 1: Assume that φ is a unital isomorphism and A = Z. In this casethe skew Laurent polynomial ring is the crossed product by Z; R[t+, t−, φ] =R[t, t−1, φ]. Let i± : R → R[t±, φ] and j± : R[t±, φ] → R[t+, t−, φ] be theinclusion maps. By the proof of [39, Theorem 2.1], there is a homotopy fibration

K(R)⊕K(R)ψ // K(R[t+, φ])⊕K(R[t−, φ])

[j+,j−]// K(R[t+, t−, φ])

and K(R[t±, φ]) = K(R)⊕NK(R, φ)±. Here

ψ =

[i+ i+

i− i− φ

]

Application of Lemma 3.1 yields the fibration of the theorem; this finishes thecase when φ is a unital isomorphism.

Step 2: Assume that B is an H ′-unital ring and that φ : B → B is an isomor-phism. Then by the previous step, the augmentation B → Z induces a map offibration sequences

K(B)1−φ //

K(B)⊕NK(B, φ)+ ⊕NK(B, φ)− //

K(B[t+, t−, φ])

K(Z)

0// K(Z) // K(Z[t, t−1]).



Since B[t±, φ] and B[t+, t−, φ] are H ′-unital by Proposition 2.8, the fibers ofthe vertical maps give the fibration of the theorem.

Step 3: Assume that R is unital and let φ be a corner isomorphism. Let A bean H ′-unital ring. Write S = φ−1R for the colimit of the inductive system

Rφ // R

φ // Rφ // . . . .

Then φ induces an automorphism φ : S → S. Set Rn = R; then B = S ⊗A =colimnRn ⊗A is H ′-unital, since Rn ⊗A is H ′-unital by hypothesis, and H ′-

unitality is preserved under filtering colimits (see Remark 2.2). Since φ⊗ 1 isan automorphism of B, Step 2 gives

K(B[t+, t−, φ⊗ 1]) = hocofiber(1− φ⊗ 1 : K(B)→ K(B))(3.7)

⊕NK(B, φ⊗ 1)+ ⊕NK(B, φ⊗ 1)−.

Because K-theory commutes with filtering colimits, we have K(B) = (φ ⊗1)−1K(R⊗A). Thus by Lemma 3.3,

(3.8) hocofiber(1− φ⊗ 1 : K(B)→ K(B)) ∼=hocofiber(1 − φ⊗ 1 : K(R⊗A)→ K(R⊗A)).

Write ϕn : Rn → S for the canonical map of the colimit, and put en = ϕn(1).For n ≥ 0, there is a ring isomorphism ψn : (R ⊗ A)[t+, t−, φ ⊗ 1] → (en ⊗1)B[t, t−1, φ⊗1](en⊗1), where en⊗1 ∈ S⊗A, such that ψn(r⊗a) = ϕn(r)⊗a,and ψn(t+) = (en ⊗ 1)t(en ⊗ 1), and ψn(t−) = (en ⊗ 1)t−1(en ⊗ 1).

Consider the map η : (R ⊗ A)[t+, t−, φ ⊗ 1] −→ (R ⊗ A)[t+, t−, φ ⊗ 1],η(x) = t+xt−. There is a commutative diagram

(3.9)

(R⊗A)[t+, t−, φ⊗ 1]ψn−−−−→ (en ⊗ 1)B[t, t−1, φ⊗ 1](en ⊗ 1)

η

y i

y

(R⊗A)[t+, t−, φ⊗ 1]ψn+1−−−−→ (en+1 ⊗ 1)B[t, t−1, φ⊗ 1](en+1 ⊗ 1).

It follows that B[t+, t−, φ ⊗ 1] = η−1(R ⊗ A)[t+, t−, φ ⊗ 1]. Hence we have

K(B[t+, t−, φ⊗]) ∼= η−1K((R⊗A)[t+, t−, φ⊗1]). But since t−t+ = 1, the mapη induces the identity on K((R⊗A)[t+, t−, φ⊗ 1]) (e.g. by [7, 2.2.6]). Thus

(3.10) K((R⊗A)[t+, t−, φ⊗ 1]) ∼= K(B[t+, t−, φ⊗ 1]).

In addition we have

(3.11) NK(R⊗A, φ⊗ 1)± ∼= NK(B, φ⊗ 1)±.

Rewrite (3.7) using (3.8), (3.11) and (3.10) to finish the third (and final) step.



4. Matrices associated to finite quivers

Let E be a finite quiver. Write E0 for the set of vertices and E1 for the setof arrows. In this section we assume both E0 and E1 are finite, of cardinalitiese0 and e1. If α ∈ E1, we write s(α) for its source vertex and r(α) for its range.There are two matrices with non-negative integer coefficients associated withE; these are best expressed in terms of the range and source maps r, s : E1 →E0. If f : E1 → E0 is a map of finite sets, and χx, χy are the characteristicfunctions of x and y, we write

f∗ : ZE0 → ZE1 , f∗(χy) =∑

f(x)=y

χx

f∗ : ZE1 → ZE0 , f∗(χx) = χf(x).

Put

(4.1) ME = r∗s∗ N ′E = s∗r∗

We identify these homomorphisms with their matrices with respect to thecanonical basis. The matrices ME = [mα,β] ∈ Me1Z and N ′E = [ni,j ] ∈ Me0Zare given by

mα,β = δr(α),s(β)

ni,j = #α ∈ E1 : s(α) = i, r(α) = jFor i = 0, 1, we consider the chain complex Ci concentrated in degrees 0 and1, with Cij = Zei if j = 0, 1, and with boundary map 1 − N ′E if i = 0 and1−ME if i = 1. Pictorially

C0 : ZE0

1−N ′E // ZE0

C1 : ZE1

1−ME

// ZE1 .(4.2)

Lemma 4.3. The maps r∗ and s∗ induce inverse homotopy equivalences C0

C1.

Proof. Straightforward.

Proposition 4.4. If X is a spectrum, then hocofiber(1−ME : Xe1 → Xe1) ∼=hocofiber(1 −N ′E : Xe0 → Xe0).

Proof. Note r∗ induces a map

Xe01−N ′

E //

r∗

Xe0

r∗

// hocofiber(1−N ′E)

f

Xe1

1−ME // Xe1 // hocofiber(1−ME).

From the long exact sequences of homotopy groups of the fibrations above, weobtain



0

0

H0(C

0 ⊗ πn(X))r∗ //

H0(C1 ⊗ πn(X))

πn hocofiber(1−N ′E)

f //

πn hocofiber(1−ME)

H1(C

0 ⊗ πn−1X)r∗ //

H1(C1 ⊗ πn−1X)

0 0.

By Lemma 4.3, the horizontal maps at the two extremes are isomorphisms;it follows that the map in the middle is an isomorphism too.

Recall that a vertex i ∈ E0 is called a source (respectively, a sink) in caser−1(i) = ∅ (respectively, s−1(i) = ∅). We will denote by Sink(E) the sets ofsinks of E.

5. K-theory of the Leavitt algebra I: finite quivers withoutsinks

Let E be a finite quiver and M = ME . The path ring of E is the ringP = PZ(E) with one generator for each arrow α ∈ E1 and one generator pi foreach vertex i ∈ E0, subject to the following relations

pipj = δi,jpi, (i, j ∈ E0)(5.1)

ps(α)α = α = αpr(α), (α ∈ E1)(5.2)

The ring P has a basis formed by the pi, the α, and the products α1 · · ·αnwith r(αi) = s(αi+1). We think of these as paths in the quiver, of lengths, 0,1 and n, respectively. Observe that P is unital, with 1 =

∑i∈E0

pi.Consider the opposite quiver E∗; this is the quiver with the same sets of

vertices and arrows, but with the range and source functions switched. ThusE∗i = Ei (i = 0, 1) and if we write α∗ for the arrow α ∈ E1 considered asan arrow of E∗, we have r(α∗) = s(α) and s(α∗) = r(α). The path ringP ∗ = P (E∗) is generated by the pi (i ∈ E0) and the α∗ ∈ E∗1 ; the relation(5.1) is satisfied, and we also have

(5.3) pr(α)α∗ = α∗ = α∗ps(α), (α ∈ E1).

The Leavitt path ring of E is the ring L = LZ(E) on generators pi (i ∈ E0),α ∈ E1, and α∗ ∈ E∗1 , subject to relations (5.1), (5.2), and (5.3), and to the



following two additional relations

α∗β = δα,βpr(α)(5.4)

pi =∑

s(α)=i

αα∗ (i ∈ E0 \ Sink(E))(5.5)

From these last two relations we obtain

α∗α =∑

s(β)=r(α)

ββ∗

=∑

β∈E1

mβ,αββ∗.(5.6)

It also follows, in case E has no sinks, that the qβ = ββ∗ are a complete systemof orthogonal idempotents; we have

(5.7)∑

β∈E1

qβ = 1, qαqβ = δα,βqβ .

The ring L is equipped with an involution and a Z-grading. The involutionx 7→ x∗ sends α 7→ α∗ and α∗ 7→ α. The grading is determined by |α| = 1,|α∗| = −1. By [4, proof of Theorem 5.3], we have L0 =

⋃∞n=0 L0,n, where L0,n

is the linear span of all the elements of the form γν∗, where γ and ν are pathswith r(γ) = r(ν) and |γ| = |ν| = n. For each i in E0, and each n ∈ Z+, let usdenote by P (n, i) the set of paths γ in E such that |γ| = n and r(γ) = i. Thering L0,0 is isomorphic to

∏i∈E0 k. In general the ring L0,n is isomorphic to

[ n−1∏

m=0

( ∏

i∈Sink(E)

M|P (m,i)|(Z))]×[ ∏

i∈E0

M|P (n,i)|(Z)].

The transition homomorphism L0,n → L0,n+1 is the identity on the fac-tors

∏i∈Sink(E)M|P (m,i)|(Z), for 0 ≤ m ≤ n − 1, and also on the factor∏

i∈Sink(E)M|P (n,i)|(Z) of the last term of the displayed formula. The tran-

sition homomorphism∏

i∈E0\Sink(E)

M|P (n,i)|(Z)→∏

i∈E0

M|P (n+1,i)|(Z)

is a block diagonal map induced by the following identification in L(E)0: Amatrix unit in a factor M|P (n,i)|(Z), where i ∈ E0 \ Sink(E), is a monomial ofthe form γν∗, where γ and ν are paths of length n with r(γ) = r(ν) = i. Sincei is not a sink, we can enlarge the paths γ and ν using the edges that i emits,obtaining paths of length n + 1, and relation (5.5) in the definition of L(E)gives

γν∗ =∑

α∈E1|s(α)=i

(γα)(να)∗.



Assume E has no sources. For each i ∈ E0, choose an arrow αi such thatr(αi) = i. Consider the elements

t+ =∑

i∈E0

αi, t− = t∗+.

One checks that t−t+ = 1. Thus, since |t±| = ±1, the endomorphism

φ : L→ L, φ(x) = t+xt−

is homogeneous of degree 0 with respect to the Z-grading. In particular itrestricts to an endomorphism of L0. By [3, Lemma 2.4], we have

(5.8) L = L0[t+, t−, φ].

For a unital ring A, we may define the Leavitt path A-algebra LA(E) in thesame way as before, with the proviso that elements of A commute with thegenerators pi, α, α∗. Observe that

(5.9) LA(E) = LZ(E)⊗ A.If A is a not necessarily unital ring, we take (5.9) as the definition of LA(E).We may think of LZ(E) as the most basic Leavitt path ring.

Let e′0 = |Sink(E)|. We assume that E0 is ordered so that the first e′0elements of E0 correspond to its sinks. Accordingly, the first e′0 rows of thematrix N ′E are 0. Let NE be the matrix obtained by deleting these e′0 rows.The matrix that enters the computation of the K-theory of the Leavitt pathalgebra is (

01e0−e′0

)−N t

E : Ze0−e′0 −→ Ze0 .

By a slight abuse of notation, we will write 1−N tE for this matrix. Note that

1 −N tE ∈ Me0×(e0−e′0)(Z). Of course NE = N ′E in case E has no sinks, where

N ′E is introduced in Section 4.

Theorem 5.10. Let A be an H ′-unital ring, E a finite quiver, M = ME andN = NE. Assume the quiver E has no sources. We have

K(LA(E)) ∼=NK(L0 ⊗A, φ⊗ 1)+ ⊕NK(L0 ⊗A, φ⊗ 1)−

⊕ hocofiber(K(A)e0−e′0

1−Nt−→ K(A)e0).

Moreover, if in addition E has no sinks then

K(LA(E)) ∼=NK(L0 ⊗A, φ⊗ 1)+ ⊕NK(L0 ⊗A, φ⊗ 1)−

⊕ hocofiber(K(A)e11−Mt

−→ K(A)e1).

Proof. If E has no sinks, then Proposition 4.4 applied to E∗ gives

hocofiber(K(A)e11−Mt

−→ K(A)e1 ) ∼= hocofiber(K(A)e01−Nt−→ K(A)e0).

Thus it suffices to prove the first equivalence of the theorem. By (5.8),

LA(E) = (L0 ⊗A)[t+, t−, 1⊗ φ].



Note L0 ⊗ A is a filtering colimit of rings of matrices with coefficients in A.Since A is H ′-unital by hypothesis, each such matrix ring is H ′-unital, whenceL0 ⊗A is H ′-unital. Hence, by Theorem 3.6

K(LA(E)) ∼=NK(L0 ⊗A, φ⊗ 1)+ ⊕NK(L0 ⊗A, φ⊗ 1)−

⊕ hocofiber(K(L0 ⊗A)1−φ⊗1−→ K(L0 ⊗A)).

As explained in the paragraph immediately above the theorem, we have L0 =⋃∞n=0 L0,n. Since E has no sources, it follows that L0,n is the product of

exactly ne′0 + e0 = (n+ 1)e′0 + (e0 − e′0) matrix algebras; thus K(A⊗ L0,n) ∼=K(A)(n+1)e′0+(e0−e

′0), since A is H ′-unital and K-theory is matrix stable on H ′-

unital rings (by Theorem 2.6). Moreover the inclusion L0,n ⊂ L0,n+1 induces

∆n :=

(1(n+1)e′0

00 N t

): K(A)(n+1)e′0+(e0−e

′0) −→ K(A)(n+1)e′0+e0 .

Now, for a path γ on E, we have

φ(γγ∗) =∑

i,j

αiγγ∗α∗j = (αs(γ)γ)(αs(γ)γ)

∗,

so that φ⊗ 1 induces

Ωn :=

(0

1ne′0+e0

): K(A)ne

′0+e0 = K(L0,n ⊗A) −→ K(A)(n+1)e′0+e0 .

Summing up, we get a commutative diagram(5.11)

K(L0,n ⊗A)∆n−−−−→ K(L0,n+1 ⊗A) −−−−→ · · · −−−−→ K(L0 ⊗A)

∆n−Ωn

yy∆n+1−Ωn+1

y1−φ⊗1

K(L0,n+1 ⊗A)∆n+1−−−−→ K(L0,n+2 ⊗A) −−−−→ · · · −−−−→ K(L0 ⊗A).

Note that elementary row operations take ∆n − Ωn to 1(n+1)e′0⊕ (N t − 1);

hence there is an elementary matrix h such that h(∆n − Ωn) = 1(n+1)e′0⊕

(N t−1). Moreover one checks that h restricts to the identity on 0⊕K(A)e0 ⊂K(A)(n+1)e′0+e0 . It follows that the inclusion in+1 : K(A)e0 → 0 ⊕K(A)e0 ⊂K(A)(n+1)e′0+e0 induces an equivalence

C := hocofiber(K(A)e0−e′0

1−Nt−→ K(A)e0 )

∼= hocofiber(K(L0,n ⊗A)∆n−Ωn−→ K(L0,n ⊗A)),

and that furthermore, the diagram

K(L0,n ⊗A)∆n−Ωn //

Ωn

K(L0,n+1 ⊗A)

Ωn+1

// C

K(L0,n+1 ⊗ A)∆n+1−Ωn+1 // K(L0,n+2 ⊗A) // C



is homotopy commutative. Hence

K(L0,n ⊗A)∆n−Ωn //

∆n

K(L0,n+1 ⊗A)

∆n+1

// C

K(L0,n+1 ⊗A)∆n+1−Ωn+1 // K(L0,n+2 ⊗A) // C

is homotopy commutative too. Thus hocofiber(1 − 1 ⊗ φ : K(L0 ⊗ A) →K(L0 ⊗A)) ∼= C.

6. K-theory of Leavitt algebras II: row-finite quivers

A quiver E is said to be row-finite if for each i ∈ E0, the set s−1(i) = α ∈E1 | s(α) = i is finite. This is equivalent to saying that the adjacency matrixN ′E of E is a row-finite matrix. For a row-finite quiver E, the Leavitt pathalgebras LZ(E) and LA(E) are defined exactly as in the case of a finite quiver.

Recall that a complete subgraph of a quiver E is a subquiver F such thatfor every v ∈ F0 either s−1

F (v) = ∅ or s−1F (v) = s−1

E (v). If F is a completesubgraph of E, then there is a natural homomorphism LA(F ) → LA(E) (see[4, Lemma 3.2]).

Lemma 6.1. Let E be a finite quiver and let F be a subquiver of E withd = |F | and d′ = |Sink(F )|. Let A be a unital ring. Suppose there is a vertexv ∈ E0\F0 such that s−1

E (v) 6= ∅ and rE(s−1E (v)) ⊆ F0. Consider the subquiver

F ′ of E with F ′0 = F0 ∪ v, F ′1 = F1 ∪ s−1E (v). Then the following properties

hold:

(1) LA(F ) is a full corner in LA(F ′). In particular LA(F ) and LA(F ′)are Morita equivalent.

(2) hocofiber(1 −N tF : K(A)d−d

′ → K(A)d) ∼=hocofiber(1 −N t

F ′ : K(A)d+1−d′ → K(A)d+1).

Proof.

(1) Set p =∑

i∈F0pi ∈ LA(F ′). It is easily seen that LA(F ) ∼= pLA(F ′)p.

Since p is a full idempotent in LA(F ′), this proves (1).

(2) Recall that we write 1−N tF for the d× (d−d′)-matrix

(0

1d−d′

)−N t

F .

Note that v is a source in F ′, so for every j ∈ F ′0 we have nF′

jv = 0.The matrices

(0

1d+1−d′

)−N t

F ′ and

(

01d−d′

)−N t

F 0

0 1

are clearly equivalent by elementary transformations, from which theresult follows.



For a path γ ∈ En, with n ≥ 1, we denote by v(γ) the set of all verticesappearing as range or source vertices of the arrows of γ. If i ∈ E0 is a trivialpath, we set v(i) = i. Write LE = γ ∈ E∗ | |v(γ)| = |γ| + 1, the set ofpaths without repetitions of vertices. Denote by rE∗

and sE∗the extensions of

rE and sE respectively to the set of all paths in E.Given a quiver with oriented cycles, we define a subquiver E of E by setting

E0 = i ∈ E0 | rE∗(i) * LE and E1 = α ∈ E1 | sE(α) ∈ E0. Observe that

this is a well-defined quiver because, if sE(α) ∈ E0, then rE(α) ∈ E0 as well.

If E does not have oriented cycles, then we define E as the empty quiver.

Lemma 6.2. Let E be a quiver. Then E is a complete subgraph of E withoutsources, and if γ ∈ E∗ is a nontrivial closed path then γ ∈ E∗.

Proof. The result is clear in case E does not have oriented cycles. Supposethat E has oriented cycles. By definition, E is a complete subgraph of E.Observe that if i ∈ E0 then s−1

E∗(i) ⊆ E∗. Now if γ ∈ E∗ is a nontrivial closed

path we have s(γ) = r(γ) ∈ E0 and so γ ∈ E∗.Pick v ∈ E0. By construction there is γ = α1 · · ·αm ∈ rE∗

(v) such that|v(γ)| ≤ m. Hence there exists an index i such that there is a nontrivial closed

path based on rE(αi). Then rE(αi) ∈ E0 and so v ∈ E0. Therefore E has nosources.

We are now ready to obtain our main general result for a row-finite quiver.

Theorem 6.3. Let A be either a ring with local units or an H ′-unital ringwhich is torsion free as a Z-module, and let E be a row-finite quiver. Thenthere is a map

hocofiber(K(A)(E0\Sink(E)) 1−NtE−→ K(A)(E0))→ K(LA(E)),

which induces a naturally split monomorphism at the level of homotopy groups

(6.4) π∗(hocofiber(K(A)(E0\Sink(E)) 1−NtE−→ K(A)(E0))→ K∗(LA(E))).

Proof. We first deal with the case of a finite quiver E. Set d = |E0| andd′ = |Sink(E)|.

Consider the subquiver F of E given by F0 = E0 ∪ Sink(E) and F1 = E1.Using Lemma 6.2 we see that F is a complete subgraph of E such that everynontrivial closed path on E has all its arrows and vertices in F . Moreover wehave Sink(F ) = Sink(E).

Set p = |F0| and k = d − p. Suppose that k > 0. In this case we willbuild a chain of complete subgraphs of E, F = F 0 ⊂ F 1 ⊂ · · · ⊂ F k = E,with |F i+1

0 \ F i0| = 1, and such that the following conditions hold for everyi = 0, . . . , k − 1:

(i) Sink(F i) = Sink(E).(ii) LZ(F i) is a full corner in LZ(F i+1).



(iii)

hocofiber(

(0

1p+i−d′

)−N t

F i : K(A)p+i−d′ −→ K(A)p+i)

∼= hocofiber(

(0

1p+i+1−d′

)−N t

F i+1 : K(A)p+i+1−d′ −→ K(A)p+i+1).

Suppose we have defined F i for 0 ≤ i < k. We are going to define F i+1.We first show that there is a vertex v ∈ E0 \ F i0 such that rE(s−1

E (v)) ⊆ F i0 .

Pick v1 ∈ E0 \ F i0. Since Sink(F i) = Sink(E) we have that s−1E (v1) 6= ∅. If

there exists α1 ∈ s−1E (v1) such that rE(α1) /∈ F i0 , set v2 = rE(α1). Since the

number of vertices in E0 \F i0 is finite, proceeding in this way we will get eithera vertex v ∈ E0 \F i0 such that rE(s−1

E (v)) ⊆ F i0 or a path γ = α1α2 · · ·αm withαj ∈ E1 \F i1 such that rE(αm) ∈ rE(α1), . . . , rE(αm−1). But the latter casecannot occur: the path γ would not belong to LE and consequently we wouldobtain rE(αm) ∈ E0 ⊆ F i0 , a contradiction. Therefore we put F i+1

0 = F i0 ∪vand F i+1

1 = F i1∪s−1E (v). By construction we get (i) and that F i+1 is a complete

subgraph of E, and (ii) and (iii) follow from Lemma 6.1.Set ℓ = |v ∈ Sink(E) | r−1

E∗(v) ⊆ LE|. Then we clearly have K(LA(F )) ∼=

K(LA(E))⊕K(A)ℓ. Now by Lemma 6.2 E is a quiver without sources. Note

that |E0| − |Sink(E)| = (p − ℓ) − (d′ − ℓ) = p− d′, so from Theorem 5.10 weget a decomposition

K(LA(E)) =NK(L0(E)⊗A, φ⊗ 1)+ ⊕NK(L0(E)⊗A, φ⊗ 1)−⊕

hocofiber(

(0

1p−d′

)−N t

E: K(A)p−d

′ → K(A)p−ℓ).

Hence

K(LA(F )) ∼= K(LA(E))⊕K(A)ℓ(6.5)

∼= NK(L0(E)⊗A, φ ⊗ 1)+ ⊕NK(L0(E)⊗A, φ⊗ 1)−

⊕ hocofiber(

(0

1p−d′

)−N t

F : K(A)p−d′ → K(A)p).

This gives the result for F 0 = F . Applying inductively (ii) and (iii) to thequivers of the chain F = F 0 ⊂ F 1 ⊂ · · · ⊂ F k = E, and using Lemma 2.7,we get the assertions of theorem for finite E. Let E be a row-finite quiver.By [4, Lemma 3.2], E is the filtered colimit of its finite complete subgraphs.Since filtered colimits are exact, hocofiber commutes with them, so we get themonomorphism in (6.4). To compute the cokernel of this map, note that the

construction of the graph E is functorial in the category of row-finite quiversand complete graph homomorphisms. Moreover we get E = colim F , where Franges on the family of all finite complete subquivers of E. For each i ∈ E0 weselect an arrow αi ∈ E1 such that r(αi) = i. This choice induces a compatible

choice of arrows in the quivers F corresponding to finite complete subquiversF of E. Hence, if F 1 ⊆ F 2 are two finite complete subquivers of E, then the



corresponding corner-isomorphisms φi on L(F i)0 satisfy that φ2|L(F 1)0= φ1,

and thus we obtain maps

κ± : NK(L(F 1)0 ⊗A, φ1 ⊗ 1)± −→ NK(L(F 2)0 ⊗A, φ2 ⊗ 1)±

such that the map K(LA(F 1))→ K(LA(F 2)), written in terms of the decom-position given in Theorem 5.10, is of the form κ+⊕κ−⊕κ, where κ is the mapbetween the corresponding hocofiber terms. The result follows.

Remark 6.6. The proof above shows that cokernel of the map (6.4) can beexpressed in terms of twisted nil-K-groups. If E is finite, the cokernel isNK∗(L0(E)⊗A, φ⊗1)+⊕NK∗(L0(E)⊗A, φ⊗1)+, by (6.5). In the general case,it is the colimit of the cokernels corresponding to each of the finite completesubquivers.

7. Leavitt rings with regular supercoherent coefficients

In this section we will determine the K-theory of the Leavitt path ring of arow-finite quiver over a regular supercoherent ring k.

Recall that a unital ring R is said to be coherent if its finitely presentedmodules form an abelian subcategory of the category of all modules. We saythat R is regular coherent if it is coherent and in addition any finitely presentedmodule has finite projective dimension. Equivalently R is regular coherent ifany finitely presented module has a finite resolution by finitely generated pro-jective modules. The ring R is called supercoherent in case all polynomial ringsR[t1, . . . , tp] are coherent, see [18]. Note that every Noetherian ring is super-coherent. A more general version of regularity was introduced by Vogel, see[5]. We will call this concept Vogel-regularity. For a coherent ring R, Vogel-regularity agrees with regularity ([5, Proposition 10]). Since Vogel-regularityis stable under the formation of polynomial rings ([5, Proposition 5(3)]), itfollows that R[t1, . . . , tp] is regular for every p in case R is regular supercoher-ent. Observe also that any flat universal localization R → RΣ−1 of a regular(super)coherent ring is also regular (super)coherent. This is due to the factthat every finitely presented RΣ−1-module is induced from a finitely presentedR-module ([28, Corollary 4.5]). In particular all the rings R[t1, t

−11 , . . . , tp, t

−1p ]

are regular supercoherent if R is regular supercoherent.Next we will compute the K-theory of the Leavitt algebra of a quiver E

over a regular supercoherent coefficient ring k. As a first step, we consider thecase where E is finite and without sources.

Proposition 7.1. Let E be a finite quiver without sources and let k be a regularsupercoherent ring. Let B = φ−1L0, where L0 is the homogeneous componentof degree 0 of Lk(E). Let D = B ⊕ k be the k-unitization of B. Then D isregular supercoherent.

Proof. Since the ring corresponding to k[t1, . . . , tp] is D[t1, . . . , tp], it sufficesto show that D is regular coherent whenever k is so.



We are going to apply [18, Proposition 1.6]: If R = colimi∈I Ri, where I isa filtering poset, the ring R is a flat left Ri-module for all i ∈ I, and each Riis regular coherent, then R is regular coherent.

We will show that L0 is flat as a left L0,n-module. It is enough to show thatL0,n+1 is flat over L0,n. Observe that

L0,n+1 =⊕

|γ|≤n,r(γ)∈Sink(E)

L0,nγγ∗⊕ ⊕

|γ|=n+1

L0,n+1γγ∗,

so that we only need to analyze the terms L0,n+1γγ∗ with γ ∈ En+1. Write

γ = γ0α with γ0 ∈ En and α ∈ E1. For v ∈ E0 set

Zv,n = β ∈ E1 | r(β) = v and there exists η ∈ En such that r(η) = s(β).For each β ∈ Zv,n, select ηβ ∈ En such that r(ηβ) = s(β). Then

L0,n+1γγ∗ =

⊕

β∈Zr(α),n

L0,nηββα∗(γ0)

∗ ∼=⊕

β∈Zr(α),n

L0,nηβ(ηβ)∗.

Thus L0,n+1 is indeed projective as a L0,n-module.By [18, Proposition 1.6] we get that L0 is regular coherent. Now observe

that D = colim(eiBei ⊕ k), where ei is the image of 1 ∈ L0 through thecanonical map ϕi : L0 → B to the colimit. Since eiBei ∼= L0 is unital, we getthat eiBei ⊕ k ∼= L0 × k, where L0 × k denotes the ring direct product of L0

and k, and so it is regular coherent by the above. By another application of[18, Proposition 1.6], it suffices to check that ei+1Bei+1 ⊕ k is flat as a lefteiBei ⊕ k-module, which in turn is equivalent to checking that L0 is flat as aleft (1 − e)k × eL0e-module, where e = φ(1) =

∑i∈E0

αiα∗i . Recall that, for

i ∈ E0, αi ∈ E1 is such that r(αi) = i. We have L0 = (1 − e)L0 ⊕ eL0 andsince (1− e)L0 is flat as a left (1− e)k-module, it suffices to show that eL0 isflat as a left eL0e-module. Because

L0,1∼= kSink(E) ×

∏

i∈E0

M|P (1,i)|(Z)

we see that there is a central idempotent z in L0 such that e ∈ zL0 and e isa full idempotent in zL0, that is zL0 = zL0eL0. Now a standard argumentshows that eL0 is indeed projective as a left eL0e-module. Indeed there existsn ≥ 1 and a finitely generated projective L0-module P such that

zL0 ⊕ P ∼= (L0e)n;

tensoring this with eL0 we get eL0⊕eP ∼= (eL0e)n, as wanted. This concludes

the proof.

Our next lemma follows essentially from Waldhausen [34].

Lemma 7.2. Let R be a regular supercoherent ring and let φ be an auto-morphism of R. Extend φ to an automorphism of R[t1, t

−11 , . . . , tp, t

−1p ] by

φ(ti) = ti. Then NKn(R[t1, t−11 , . . . , tp, t

−1p ], φ)± = 0 for every p ≥ 0 and

every n ∈ Z.



Proof. For n ≥ 1 this follows from [34, Theorem 4], because, as we observedbefore, R[t1, t

−11 , . . . , tp, t

−1p ] is regular coherent. Let n ≤ 1 and assume that

NKi(R[t1, t−11 , . . . , tp, t

−1p ], φ)+ = 0 for every p ≥ 0, for every i ≥ n, and for

every automorphism φ of R. To show the result for NKn−1, it will be enoughto show that NKn−1(R, φ)+ = 0. Since R[t, t−1] is regular supercoherent wehave

Kn((R[t, t−1])[s, φ]) = Kn(R[t, t−1])⊕NKn(R[t, t−1], φ) = Kn(R[t, t−1]),

by induction hypothesis. It follows that

(7.3) Kn(R[t, t−1][s, φ]) = Kn(R)⊕Kn−1(R)

because NKn(R) = 0 again by induction hypothesis. On the other hand wehave

Kn(R[s, φ][t, t−1]) = Kn(R[s, φ])⊕Kn−1(R[s, φ])⊕NKn(R[s, φ])2(7.4)

= Kn(R)⊕Kn−1(R)⊕NKn−1(R, φ)+ ⊕NKn(R[s, φ])2.

Comparison of (7.3) and (7.4) gives

NKn−1(R, φ)+ = 0 = NKn(R[s, φ]),

as desired.

Proposition 7.5. Let k be a regular supercoherent ring and let E be a finitequiver without sources. Set d = |E0| and d′ = |Sink(E)|. Then

K(Lk(E)) ∼= hocofiber(K(k)d−d′ 1−NtE−→ K(k)d).

Proof. Let B = φ−1L0, where φ = φ ⊗ 1: L0 = LZ0 ⊗ k → L0 = LZ

0 ⊗ k isthe corner-isomorphism defined by φ(x) = t+xt−, as in Section 5. Note that

since k is regular supercoherent and B is H ′-unital we have NK(B, φ)± =

NK(B ⊕ k, φ)±, where B ⊕ k denotes the k-unitization of B. Now it followsfrom Proposition 7.1 that B⊕k is regular supercoherent. Therefore Lemma 7.2

gives that NK(B ⊕ k, φ)± = 0. It follows that NK(L0, φ)± = NK(B, φ)± =

NK(B ⊕ k, φ)± = 0 and so the result follows from Theorem 5.10.

Theorem 7.6. Let k be a regular supercoherent ring and let E be a row-finitequiver. Then

K(Lk(E)) ∼= hocofiber(K(k)(E0\Sink(E)) 1−NtE−→ K(k)(E0)).

It follows that there is a long exact sequence

Kn(k)(E0\Sink(E)) 1−NtE−→ Kn(k)

(E0)

−→ Kn(Lk(E)) −→ Kn−1(k)(E0\Sink(E)).

Proof. The case when E is finite follows from Proposition 7.5 and the argumentof the proof of Theorem 6.3. The general case follows from the finite case, bythe same argument as that given for the proof of 6.3.



Corollary 7.7. Let k be a principal ideal domain and let E be a row-finitequiver. Then

K0(Lk(E)) ∼= coker (1−N tE : Z(E0\Sink(E)) −→ Z(E0)),

and

K1(Lk(E))∼= coker (1−N tE : K1(k)

(E0\Sink(E)) −→ K1(k)(E0))

⊕ker (1 −N t

E : Z(E0\Sink(E)) −→ Z(E0)).

Remark 7.8. If we only assume that k is regular coherent in Theorem 7.6, thenthe long exact sequence in the statement terminates at K0(Lk(E)), althoughconjecturally the long exact sequence should still stand under this weaker hy-pothesis on k, see [5].

8. Homotopy algebraic K-theory of the Leavitt algebra

Homotopy algebraic K-theory, introduced by C. Weibel in [37], is a partic-ularly well-behaved variant of algebraic K-theory: it is polynomial homotopyinvariant, excisive, Morita invariant, and preserves filtering colimits. There isa comparison map

(8.1) K∗(A)→ KH∗(A).

It is proved in [37] that if A is unital and Kn(A) → Kn(A[t1, . . . , tp]) is anisomorphism for all p ≥ 1 (i.e. A is Kn-regular) then (8.1) is an isomorphismfor ∗ ≤ n. In particular if A is unital and K-regular, that is, if it is Kn-regularfor all n, then (8.1) is an isomorphism for all ∗ ∈ Z. Further, we have:

Lemma 8.2. Let A be a H ′-unital ring, torsion free as a Z-module. If A isKn-regular, then Km(A)→ KHm(A) is an isomorphism for all m ≤ n.

Proof. By Remark 2.2, A[t1, . . . , tp] is H ′-unital for all p. Hence the split exactsequence of rings

0→ A[t1, . . . , tp]→ A[t1, . . . , tp]→ Z[t1, . . . , tp]→ 0

induces a decomposition K∗(A[t1, . . . , tp]) = K∗(Z) ⊕ K∗(A[t1, . . . , tp]), since

Z is K-regular. Thus A is Kn-regular, and therefore Km(A) = KHm(A) =KHm(A) ⊕Km(Z) for m ≤ n. Splitting off the summand Km(Z), we get theresult.

Example 8.3. Examples of K-regular rings include regular supercoherent rings(see [34, Theorem 4]), and both stable and commutative C∗-algebras (see [27,3.4, 3.5] and [17, 5.3]). A theorem of Vorst (see [33]) says that if a unital ring Ris Kn-regular, then it is Km-regular for all m ≤ n. If R is commutative unitaland of finite type over a field of characteristic zero, then R is K−dimR-regular([9]).



Theorem 8.4. Let R be a unital ring and let A be a ring. Let φ : R → pRpbe a corner-isomorphism. Then

KH((R⊗A)[t+, t−, φ⊗ 1]) ∼= hocofiber(KH(R⊗A)1−φ⊗1−→ KH(R⊗A)).

Proof. We shall assume that A = Z and φ is an isomorphism; the general casefollows from this by the same argument as in the proof of Theorem 3.6, keep-ing in mind that KH satisfies excision for all (not necessarily H ′-unital) rings.By [10, Thm. 6.6.2] there exist a triangulated category kk and a functorj : Rings → kk which is matrix invariant and polynomial homotopy invari-ant, sends short exact sequences of rings to exact triangles, and is universalinitial among all such functors. Hence the functor Rings → Ho(Spectra),A 7→ KH(A), factors through an exact functor KH : kk → Ho(Spectra). By[10, Thm. 7.4.1], there is an exact triangle in kk

R1−φ // R // R[t, t−1, φ] // ΣR.

Applying KH we get an exact triangle

KH(R)1−φ // KH(R) // KH(R[t, t−1, φ]) // ΣKH(R) .

Lemma 8.5. Let R be a unital ring, e ∈ R an idempotent. Assume e is full.Further let A be any ring. Then the inclusion map eRe⊗A→ R⊗A inducesan equivalence KH(eRe⊗A)→ KH(R⊗A).

Proof. By definition, KH(R) = |[n]→ K(R[t0, . . . , tn]/ < t0 + · · ·+ tn−1 >)|.The case A = Z follows from 2.7 applied to each of the polynomial ringsR[t0, . . . , tn]/ < t0 + · · ·+ tn − 1 >. As in the proof of Lemma 2.7, the generalcase follows from the case A = Z by excision.

Theorem 8.6. Let A be a ring, and E a row-finite quiver. Then

KH(LA(E)) ∼= hocofiber(KH(A)(E0\Sink(E)) 1−NtE−→ KH(A)(E0)).

Proof. The case when E is finite and has no sources follows from Theorem 8.4using the argument of the proof of Theorem 5.10. The case for arbitrary finiteE follows as in the proof of Theorem 6.3, substituting Lemma 8.5 for 2.7. Thegeneral case follows from the finite case by the same argument as in 6.3.

Example 8.7. As an application of the theorem above, consider the case whenE is the quiver with one vertex and n+ 1 loops. In this case, LZ(E) = L1,n isthe classical Leavitt ring [24], and N t

E = [n + 1]. Hence by Theorem 8.4, weget that KH(A⊗ L1,n) is KH with Z/n-coefficients:

(8.8) KH∗(A⊗ L1,n) = KH∗(A,Z/n).

Thus the effect on KH of tensoring with L1,n is similar to the effect on Ktop

of tensoring a C∗-algebra with the Cuntz algebra On+1 ([12], [13]). If A is a



Z[1/n]-algebra, then KH∗(A,Z/n) = K∗(A,Z/n) [37, 1.6], so we may substi-tute K-theory for homotopy K-theory in the right hand side of (8.8).

9. Comparison with the K-theory of Cuntz-Krieger algebras

In this section we consider the Cuntz-Krieger C∗-algebra C∗(E) associatedto a row-finite quiver E. If A is any C∗-algebra, we write C∗A(E) = C∗(E)⊗A

for the C∗-algebra tensor product. Since C∗(E) is nuclear, there is no ambi-guity on the C∗-norm we are using here. Define a map γA

n = γAn (E) so that

the following diagram commutes

Kn(C∗A(E)) // KHn(C

∗A(E))

Kn(LA(E))

OO

γA

n // Ktopn (C∗A(E)).

The purpose of this section is to analyze when the map γAn is an isomorphism.

The following is the spectrum-level version of a result of Cuntz and Krieger[15], [14], later generalized by others; see e.g. [26, Theorem 3.2].

Theorem 9.1. Let A be a C∗-algebra and E a row-finite quiver. Then

Ktop(C∗A(E)) = hocofiber( Ktop(A)(E0\SinkE)1−NtE // Ktop(A)(E0) ).

Proof. The proof follows the same steps as the one of Theorem 8.6. In partic-ular, the same arguments allow us to reduce to the case of a finite quiver Ewith no sources. In this case essentially the same proof as in [14, Proposition3.1] applies. Namely, note that LA(E) is isomorphic to a dense ∗-subalgebraof C∗A(E), and let F be the norm completion of L0(E) ⊗ A in C∗A(E). Then

K⊗C∗A(E) is a crossed product of K⊗F by an automorphism φ, and Pimsner-Voiculescu gives an exact triangle

K⊗F 1−φ−−−−→ K⊗F −−−−→ K⊗C∗A(E) −−−−→ Σ(K⊗F)

in KK. Now stability gives the following exact triangle in KK:

(9.2) F 1−φ−−−−→ F −−−−→ C∗A(E) −−−−→ ΣFwhere φ is just a corner-isomorphism. Since C∗-alg −→ KK is universalamongst all stable, homotopy invariant, half-exact for cpc-split extensions func-tors to a triangulated category and

C∗-alg −→ Ho(Spectra), A 7→ Ktop(A)

is one such functor which in addition maps mapping cone triangles to exacttriangles in Ho(Spectra), the exact triangle (9.2) is exact in Ho(Spectra); see



[16, Theorem 8.27]. But just as in the proof of Theorem 5.10, we get

hocofiber( Ktop(F)1−φ // Ktop(F) )

∼= hocofiber( Ktop(A)(E0\SinkE)1−NtE // Ktop(A)(E0) ),

This concludes the proof.

Corollary 9.3. Assume K∗(A)→ Ktop∗ (A) is an isomorphism for ∗ = n, n−

1. Then γAn is a split surjection. If in addition K∗(A) → KH∗(A) and

K∗(LA(E)) → KH∗(LA(E)) are isomorphisms for ∗ = n, n − 1, then γn isan isomorphism.

Proof. We have

πn(hocofiber( K(A)(E0\SinkE)

1−NtE// K(A)(E0) ))

∼= πn(hocofiber( Ktop(A)(E0\SinkE)

1−NtE// Ktop(A)(E0) ))

by the five lemma. Next apply Theorems 6.3 and 9.1 to obtain the first asser-tion. For the second assertion, use Theorem 8.6.

Theorem 9.4. Let E be a finite quiver without sinks. Assume that det(1 −N tE) 6= 0. Then γC

n is an isomorphism for n ≥ 0 and the zero map for n ≤ −1.

Proof. Because C is regular supercoherent, we have

(9.5) K(LC(E)) ∼= hocofiber( K(C)(E0)1−NtE // K(C)(E0) ),

by Theorem 7.6. Thus Kn(LC(E)) = 0 for n ≤ −1, and γC0 is an isomorphism

by the five lemma. Moreover if n = | det(1−N tE)|, then n2K∗(LC(E)) = 0, by

(9.5). Hence the sequence

(9.6) 0→ Km(LC(E))→ Km(LC(E),Z/n2)→ Km−1(LC(E))→ 0

is exact for all m. On the other hand, by (9.5) and Theorem 9.1, we have a



map of exact sequences (m ∈ Z)

Km(C,Z/n2)(E0)

// Ktopm (C,Z/n2)(E0)

Km(C,Z/n2)(E0) //

Ktopm (C,Z/n2)(E0)

Km(LC(E),Z/n2) //

Ktopm (C∗C(E),Z/n2)

Km−1(C,Z/n2)(E0) //

Ktopm−1(C,Z/n

2)(E0)

Km−1(C,Z/n2)(E0) // Ktop

m−1(C,Z/n2)(E0).

By a theorem of Suslin [31] the comparison map Km(C,Z/q)→ Ktopm (C,Z/q)

is an isomorphism for m ≥ 0 and q ≥ 1. Hence the map K∗(LC(E),Z/q) →Ktop∗ (LC(E),Z/q) is an isomorphism, by Theorems 7.6 and 9.1. Combine this

together with (9.6) and induction to finish the proof.

Remark 9.7. Chris Smith, a student of Gene Abrams, has given a geometriccharacterization of those finite quivers E with no sinks which satisfy det(1 −N tE) 6= 0 [29].

Example 9.8. It follows from the theorem above that the map γAn is an isomor-

phism for every finite dimensional C∗-algebra A. Let An → An+1n be aninductive system of finite dimensional C∗-algebras; write A and A for its alge-braic and its C∗-colimit. Because K-theory commutes with algebraic filteringcolimits and Ktop commutes with C∗-filtering colimits, we conclude that, for Eas in the theorem above, the mapK∗(LA(E))→ K∗(LA(E)) is an isomorphismfor ∗ ≥ 0.

Remark 9.9. Let E be a finite quiver with sinks, E ⊂ E as in Lemma 6.2,and F = E ∪ Sink(E). Then, by Theorem 7.6 and the proof of Theorem 6.3,

Kn(LC(E) = Kn(LC(F )) = Kn(LC(E))⊕Kn(C)Sink(E). Similarly,

Ktopn (C∗C(E)) = Ktop

n (C∗C(E))⊕Ktopn (C)Sink(E).

By naturality, γCn restricts on Kn(C)Sink(E) to the direct sum of copies of the

comparison map Kn(C) → Ktopn (C). Since the latter map is not an isomor-

phism for n 6= 0, it follows that γCn is not an isomorphism either.

Remark 9.10. It has been shown that if A is a properly infinite C∗-algebrathen the comparison map K∗(A) → Ktop

∗ (A) is an isomorphism [8]. Thus

K∗(C∗C(E)) → Ktop

∗ (C∗C(E)) is an isomorphism whenever C∗C(E) is properlyinfinite.



The following proposition is a variant of a theorem of Higson (see [27, 3.4])that asserts that stable C∗-algebras are K-regular.

Proposition 9.11. Let A be an H ′-unital ring, and B a stable C∗-algebra.Then A⊗B is K-regular.

Proof. By Lemma 2.3 we may assume that A is a Q-algebra. Since A→ A[t]preserves H-unitality, the proposition amounts to showing that the functorA 7→ K∗(A⊗B) is invariant under polynomial homotopy. Observe that if A isany C∗-algebra, then A ⊗ (B⊗A) is H-unital, which implies that the functorA 7→ E(A) = K∗(A ⊗ (B⊗A)), which is stable (because K-theory is matrixstable on H ′-unital rings), is also split exact. Hence E is invariant undercontinuous homotopies, by Higson’s homotopy invariance theorem [20]. ThusE sends all the evaluation maps evt : C[0, 1]→ C to the same map. But sincethe evaluation maps evi : A[t] → A factor through evi : A ⊗ C[0, 1] → A, itfollows that A 7→ E(C) = K∗(A⊗B) is invariant under polynomial homotopies,as we had to prove.

Corollary 9.12. If B is a stable C∗-algebra and E a row-finite quiver, thenboth B and LB(E) are K-regular, and the map of Theorem 6.3

hocofiber(K(B)(E0\Sink(E)) 1−NtE−→ K(B)(E0))→ K(LB(E))

is an equivalence.

Proof. That B and LB(E) areK-regular is immediate from the proposition; byCorollary 2.4, they are also H-unital. It follows from this and from Lemma 8.2that the comparison maps K(B) → KH(B) and K(LB(E)) → KH(LB(E))are equivalences. Now apply Theorem 8.6.

Theorem 9.13. If B is a stable C∗-algebra then the map γBn is an isomor-

phism for every n and every row-finite quiver E.

Proof. The theorem is immediate from Corollary 9.12, Theorem 9.1, and thefact (proved in [21] for n ≤ 0 and in [32] for n ≥ 1) that the map Kn(B) →Ktopn (B) is an isomorphism for all n.

Remark 9.14. If B is stable, then C∗B(E) is stable, and thus the comparison

map K∗(C∗B(E)) → Ktop

∗ (C∗B(E)) is an isomorphism. Moreover we also have

KH∗(C∗B(E)) ∼= Ktop

∗ (C∗B(E)), by 9.11.

Acknowledgement. Part of the research for this article was carried out duringvisits of the third named author to the Centre de Recerca Matematica and theDepartament de Matematiques of the Universitat Autonoma de Barcelona. Heis indebted to these institutions for their hospitality.

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Received March 2, 2009; accepted June 19, 2009

Pere Ara and Miquel BrustengaDepartament de MatematiquesUniversitat Autonoma de Barcelona08193 Bellaterra (Barcelona), SpainE-mail: para,[email protected]

Guillermo CortinasDep. Matematica, Ciudad Universitaria Pab 11428 Buenos Aires, ArgentinaE-mail: [email protected]

URL: http://mate.dm.uba.ar/~gcorti




Strongly self-absorbing C∗-algebras which

contain a nontrivial projection

Marius Dadarlat and Mikael Rørdam



Abstract. It is shown that a strongly self-absorbing C∗-algebra is of real rank zero andabsorbs the Jiang-Su algebra if it contains a nontrivial projection. We also consider caseswhere the UCT is automatic for strongly self-absorbing C∗-algebras, and K-theoretical ways

of characterizing when Kirchberg algebras are strongly self-absorbing.

1. Introduction

Strongly self-absorbing C∗-algebras were first systematically studied by Tomsand Winter in [11]. The classification program of Elliott had prior to that beenseen to work out particularly well for those (separable, nuclear) C∗-algebrasthat tensorially absorb one of the Cuntz algebras O2, O∞, or the Jiang-Sualgebra Z. More precisely, thanks to deep theorems of Kirchberg, the classifi-cation of separable, nuclear, stable C∗-algebras that absorb the Cuntz algebraO2 is complete (the invariant is the primitive ideal space); and separable, nu-clear, stable C∗-algebras that absorb the Cuntz algebra O∞ are classified byan ideal related KK -theory. The situation for separable, nuclear C∗-algebrasthat absorb the Jiang-Su algebra is at present very promising (see for example[13]) but not as complete as in the purely infinite case.

The C∗-algebras O2, O∞ and Z are all examples of strongly self-absorbingC∗-algebras. They are in [11] defined to be those unital separable C∗-algebrasD 6= C for which there is an isomorphism D → D ⊗D that is approximatelyunitarily equivalent to the ∗-homomorphism d 7→ d⊗1. Strongly self-absorbingC∗-algebras are automatically simple and nuclear, and they have at most onetracial state. It is shown in [11] that if D is a strongly self-absorbing C∗-alge-bra in the UCT class, then it has the same K-theory as one of the C∗-algebras

M.D. was partially supported by NSF grant #DMS-0801173 and M.R. was supported bya grant from the Danish Natural Science Research Council (FNU).

36 Marius Dadarlat and Mikael Rørdam

in the following list: Z, UHF-algebras of infinite type, O∞, O∞ tensor a UHF-algebra of infinite type, or O2. It is an open problem if nuclear C∗-algebrasalways satisfy the UCT (and also if strongly self-absorbing C∗-algebras enjoythis property); and it is an intriguing problem, very much related to the Elliottclassification program, if the list above exhausts all strongly self-absorbingC∗-algebras. Should the latter be the case, then it would in particular followthat every strongly self-absorbing C∗-algebra absorbs the Jiang-Su algebraZ. By the Kirchberg-Phillips classification theorem, a strongly self-absorbingKirchberg algebra belongs to the list above if and only if it belongs to the UCTclass. Let us also remind the reader that a strongly self-absorbing C∗-algebrais a Kirchberg algebra if and only if it is not stably finite (or, equivalently, ifand only if it is traceless).

In Section 2 of this paper we show that every strongly self-absorbing C∗-al-gebra which contains a nontrivial projection is of real rank zero and absorbsthe Jiang-Su algebra. In Section 3 we consider K-theoretical conditions onstrongly self-absorbing Kirchberg algebras. One such condition (phrased atthe level of K-homology) characterizes the Kirchberg algebra O∞, and otherresults in Section 3 give K-theoretical characterizations on when a Kirchbergalgebra is strongly self-absorbing.

2. Strongly self-absorbing C∗-algebras with a nontrivialprojection

In this section we show that any strongly self-absorbing C∗-algebra thatcontains a nontrivial projection is automatically approximately divisible, ofreal rank zero, and absorbs the Jiang-Su algebra Z.

Lemma 2.1. There is a unital ∗-homomorphism from M3 ⊕M2 into a unitalC∗-algebra A if and only if A contains projections e, e′ such that e ⊥ e′, e ∼ e′,and 1− e− e′ - e.

Proof. It is easy to see that such projections e and e′ exist in M3 ⊕M2 andhence in any unital C∗-algebraA that is the target of a unital ∗-homomorphismfrom M3 ⊕M2.

Assume now that such projections e and e′ exist. Let v ∈ A be a partialisometry such that v∗v = e and vv∗ = e′. Put f0 = 1 − e − e′. Find asubprojection f1 of e which is equivalent to f0, and put f2 = vf1v

∗. Putg1 = e− f1 and put g2 = e′− f2 = vg1v

∗. The projections f0, f1, f2, g1, g2 thensatisfy

1 = f0 + f1 + f2 + g1 + g2, f0 ∼ f1 ∼ f2, g1 ∼ g2.

Extending the sets f0, f1, f2 and g1, g2 to sets of matrix units for M3 andM2, respectively, yields a unital ∗-homomorphism from M3 ⊕M2 into A. (Ifthe fj ’s are zero or if the gj ’s are zero, then this ∗-homomorphism will fail tobe injective, and will instead give a unital embedding of M2 or M3 into A.)


Strongly self-absorbing C∗-algebras 37

If D is any unital (nuclear1) C∗-algebra then we let D⊗n denote the n-foldtensor product D ⊗ D ⊗ · · · ⊗ D (with n tensor factors), and we let D⊗∞

denote the infinite tensor product⊗∞

n=1D. The latter is the inductive limitof the sequence

D → D⊗2 → D⊗3 → D⊗4 → · · · ,(with connecting mappings d 7→ d ⊗ 1D). We shall view D as a (unital) sub-C∗-algebra of D⊗n, D⊗n as a sub-C∗-algebra of D⊗m (if n ≤ m), and finallyD and D⊗n are viewed as subalgebras of D⊗∞.

If x ∈ D⊗n, then x⊗k will denote the k-fold tensor product

x⊗k = x⊗ x⊗ x⊗ · · · ⊗ x ∈ D⊗kn.The proof of the lemma below resembles the proof of [9, Lemma 6.4].

Lemma 2.2. Let D be a strongly self-absorbing C∗-algebra, and let p be aprojection in D. Consider the following projections in D ⊗D,

e1 = p⊗ (1 − p), e′1 = (1− p)⊗ p, f = p⊗ p+ (1− p)⊗ (1− p).For each natural number n consider also the following projections in D⊗2(n+1),

en+1 = f⊗n ⊗ p⊗ (1− p), e′n+1 = f⊗n ⊗ (1− p)⊗ p.It follows that the projections e1, e2, . . . , e

′1, e′2, . . . are pairwise orthogonal in

D⊗∞, and that ej ∼ e′j. Moreover, for each natural number n, set

En = e1 + e2 + · · ·+ en, E′n = e′1 + e′2 + · · ·+ e′n.

Then En ⊥ E′n, En ∼ E′n, and

(1) 1− En − E′n = f⊗n.

Proof. The equivalence ej ∼ e′j comes from the fact that the flip automorphisma⊗b 7→ b⊗a onD⊗D is approximately inner whenD is strongly self-absorbing.The projections e1, e2, . . . , e

′1, e′2, . . . are pairwise orthogonal by construction.

The only thing left to prove is (1). We prove this by induction after n, andnote first that (1) for n = 1 follows from the fact that e1 +e′1 +f = 1. Supposethat (1) holds for some n ≥ 1. Then

1− En+1 − E′n+1

= 1− En − E′n − en+1 − e′n+1

= f⊗n ⊗ 1D ⊗ 1D − f⊗n ⊗ p⊗ (1− p)− f⊗n ⊗ (1− p)⊗ p= f⊗(n+1).

Lemma 2.3. Let D be a strongly self-absorbing C∗-algebra and let p be aprojection in D such that p 6= 1. Then there exists a natural number n suchthat p⊗n - 1− p⊗n in D⊗n.

1We shall here exclusively be concerned with nuclear C∗-algebras, where the tensor prod-uct is unique; otherwise we must specify a tensor product, for example the minimal one.



Proof. To simplify the notation we express our calculations in terms of themonoid V (D) of Murray-von Neumann equivalence classes of projections in Dand in matrix algebras over D. Let [e] ∈ V (D) denote the equivalence classcontaining the projection e in (a matrix algebra over) D.

Since D is simple and p 6= 1 there is a natural number n such that n[1−p] ≥[p]. It follows that

[1 − p⊗n] ≥ [(1− p)⊗ p⊗ · · · ⊗ p] + [p⊗ (1− p)⊗ · · · ⊗ p]+ [p⊗ p⊗ · · · ⊗ (1− p)]

= n[(1− p)⊗ p⊗ · · · ⊗ p]≥ [p⊗ p⊗ p⊗ · · · ⊗ p]= [p⊗n],

where the equality between the second and third expression holds because theflip on a strongly self-absorbing C∗-algebra is approximately inner.

Lemma 2.4. Let D be a strongly self-absorbing C∗-algebra, let p be a projec-tion in D⊗k, and let e be a projection in D⊗l for some natural numbers k andl. Assume that p 6= 1 and that e 6= 0. It follows that there exists a naturalnumber n such that p⊗n - e in D⊗∞.

Proof. Let d be a natural number such that dk ≥ l. Upon replacing p with

p⊗d, e with e⊗ 1⊗(dk−l)D , and D with D⊗dk we can assume that p and e both

belong to D. Use Lemma 2.3 to find m such that p⊗m - 1−p⊗m. By replacing

p with p⊗m, e with e⊗ 1⊗(m−1)D , and D with D⊗m we can assume that p and

e both belong to D and that p - 1− p.Now, p ∼ q ≤ 1 − p for some projection q in D. In the language of the

monoid V (D) we have

[1D⊗k] ≥ [(p+ q)⊗k] = 2k[p⊗k]

for any natural number k. Using simplicity of D we can choose n such that2n−1[e] ≥ [p]. Then

[e] = [e⊗ 1D⊗(n−1)] ≥ 2n−1[e⊗ p⊗(n−1)] ≥ [p⊗n],

in V (D⊗n) as desired, where we in the first identity have used that the em-

bedding of D into D⊗n maps e onto e⊗ 1D⊗(n−1).

Theorem 2.5. Let D be a strongly self-absorbing C∗-algebra. Then the fol-lowing three conditions are equivalent:

(1) D contains a nontrivial projection (i.e., a projection other than 0and 1).

(2) D is approximately divisible.(3) D is of real rank zero.

If any of the three equivalent conditions are satisfied, then D absorbs the Jiang-Su algebra, i.e., D ∼= D ⊗Z.



Proof. (i)⇒ (ii). IfD is strongly self-absorbing, then there is an asymptoticallycentral sequence of embeddings of D into itself, i.e., a sequence ρk : D → D ofunital ∗-homomorphisms such that ‖ρk(x)y − yρk(x)‖ → 0 as k → ∞ for allx, y ∈ D.

Identify D with D⊗∞0 where D0∼= D. Take a nontrivial projection p in

D0. For each natural number n let En, E′n ∈ D⊗2n

0 be as in Lemma 2.2(corresponding to our nontrivial projection p). Then en 6= 0, En 6= 0, and so0 6= f⊗n 6= 1. Use (1) and Lemma 2.4 to find n such that 1 − En − E′n -

p ⊗ (1 − p) ≤ En. It then follows from Lemma 2.1 that there is an injectiveunital ∗-homomorphism from M3⊗M2 into D⊗2n

0 ⊆ D. Composing this unital∗-homomorphism with the unital ∗-homomorphism ρk yields an asymptoticallycentral sequence of unital ∗-homomorphisms from M3⊗M2 into D. This showsthat D is approximately divisible.

(ii)⇒ (iii). It is shown in [2] that a simple approximately divisible C∗-alge-bra is of real rank zero if and only if projections in the C∗-algebra separate thequasitraces. As quasitraces on an exact C∗-algebra are traces, [7], a result thatapplies to our case since strongly self-absorbing C∗-algebras are nuclear andhence exact, and since a strongly self-absorbing C∗-algebra has at most onetracial state, quasitraces are automatically separated by just one projection,say the unit.

(iii) ⇒ (i). This is trivial. The only C∗-algebra of real rank zero that doesnot have a nontrivial projection is C, the algebra of complex numbers. ThisC∗-algebra is not strongly self-absorbing by convention.

Finally, any simple approximately divisible C∗-algebra is Z-absorbing, cp.[12].

Lemma 2.6. Let D be a strongly self-absorbing C∗-algebra. Then K0(D) hasa natural structure of a commutative unital ring with unit [1D]. If τ is a unitaltrace on D, then τ induces a morphism of unital rings τ∗ : K0(D)→ R.

Proof. Fix an isomorphism γ : D ⊗ D → D. The multiplication on K0(D)is defined by composing γ∗ : K0(D ⊗ D) → K0(D) with the canonical mapK0(D)⊗K0(D)→ K0(D ⊗D). Since any two unital ∗-homomorphisms fromD ⊗D to D are approximately unitarily equivalent, the above multiplicationis well-defined and commutative. We leave the rest of proof for the reader,but note that if D has a unital trace, then τ ⊗ τ is the unique unital trace ofD ⊗D.

Proposition 2.7. Let D be a strongly self-absorbing C∗-algebra. Suppose thatD is quasidiagonal and that K0(D) is torsion free. Then either K0(D) ∼= Zor there is a UHF algebra B of infinite type such that K0(D) ∼= K0(B). If, inaddition, D is assumed to contain a nontrivial projection, then D ⊗ B ∼= D,where B is as above.

Proof. Since D is quasidiagonal it embeds unitally in the universal UHF alge-bra BQ and D ⊗ BQ

∼= BQ, as explained in [5, Rem. 3.10]. The restriction of



the unital trace of BQ to D is denoted by τ . Thus we have an exact sequence

0 // H // K0(D)τ∗ // τ∗K0(D) // 0

where H is the kernel of τ∗. Since Z ⊆ τ∗K0(D) ⊆ Q, and K0(D) ⊗ Q ∼= Q,the map τ∗ ⊗ idQ : K0(D) ⊗ Q → τ∗K0(D) ⊗ Q is an isomorphism. ThereforeH ⊗ Q = 0 and so H is a torsion subgroup of K0(D). But we assumed thatK0(D) is torsion free and hence H = 0 and τ∗ : K0(D) → τ∗K0(D) ⊆ Q isan isomorphism of unital rings. The unital subrings of Q are easily determinedand well-known. They are parameterized by arbitrary sets P of prime numbers.For each P the corresponding ring RP consists of rational numbers r/s with rand s relatively prime and such that all prime factors of s are in P . If P = ∅then RP = Z, otherwise RP is isomorphic to the K0-ring associated to a UHFalgebra B of infinite type.

Suppose now that D contains a nontrivial projection. By Theorem 2.5, Dhas real rank zero and absorbs the Jiang-Su algebra Z. In particular, K0(D)is not Z and is hence isomorphic (as a scaled abelian group) to K0(B) for someUHF-algebra B of infinite type. It follows from [8] that D has stable rank oneand that K0(D) is weakly unperforated. Moreover, by [1, Sect. 6.9], K0(D)has the strict order induced by τ∗. The isomorphism K0(B) ∼= K0(D) of scaledabelian groups is therefore an order isomorphism, and by the properties of Destablished above we can conclude that B embeds unitally into D, whenceD ⊗B ∼= D.

Corollary 2.8. Let D be a strongly self-absorbing C∗-algebra with torsionfree K0-group. Suppose that D contains a nontrivial projection and that Dembeds unitally into the UHF algebra Mp∞ for some prime number p. ThenD ∼= Mp∞ .

Proof. By Proposition 2.7 there is a prime q such that Mq∞ in containedunitally in D and hence in Mp∞ . From this we deduce that q = p. Finallysince Mp∞ ⊆ D ⊆Mp∞ we conclude that D ∼= Mp∞ .

3. Strongly self-absorbing algebras and K-theory

The class of strongly self-absorbing Kirchberg algebras satisfying the UCTwas completely described in [11]. In this section we give properties and charac-terizations of strongly self-absorbing Kirchberg algebras which can be derivedwithout assuming the UCT. For a unital C∗-algebra D we denote by νD theunital ∗-homomorphism C→ D. When the C∗-algebra D is clear from contextwe will write ν instead of νD.

Proposition 3.1. Let D be a strongly self-absorbing C∗-algebra. If D is notfinite and the unital ∗-homomorphism C → D induces a surjection K0(D) →K0(C), then D ∼= O∞.

Proof. By [11, Prop. 5.12], two strongly self-absorbing C∗-algebras are isomor-phic if and only if they embed unitally into each other. Thus it suffices to show



the existence of unital ∗-homomorphisms O∞ → D and D → O∞. Since Dis not finite, it must be a Kirchberg algebra, see [11, Sec. 1], and hence O∞embeds unitally in D by [10, Prop. 4.2.3]. It remains to show that D embedsunitally in O∞.

By assumption, the map ν∗ : KK (D,C)→ KK (C,C) is surjective. By mul-tiplying with the KK -equivalence class given by the unital morphism C→ O∞,we obtain that the map ν∗ : KK (D,O∞) → KK (C,O∞) is surjective. Ifϕ : D → O∞ ⊗K is a ∗-homomorphism, then, after identifying KK (C,O∞) ∼=K0(O∞), the map ν∗ sends [ϕ] to the class [ϕ(1D)] ∈ K0(O∞). By [10,Thm. 8.3.3] each element of KK (D,O∞) is represented by a ∗-homomorphism.Therefore, by the surjectivity of ν∗, there is a ∗-homomorphism ϕ : D →O∞ ⊗ K such that [ϕ(1D)] = [1O∞

]. Since these are both full projections, by[10, Prop. 4.1.4] there is a partial isometry v ∈ O∞⊗K such that v∗v = ϕ(1D)and vv∗ = 1O∞

. Then vϕv∗ is a unital embedding D → O∞.

Remark 3.2. Note that the isomorphism D ∼= O∞ was obtained withoutassuming that D satisfies the UCT. Let us argue that assumptions of Propo-sition 3.1 are natural. Let A and B be unital C∗-algebras and let ν : C → Aand ν : C → B be the corresponding unital ∗-homomorphisms. The conditionthat there is a morphism of pointed groups (K0(A), [1A])→ (K0(B), [1B]) canbe viewed as the condition that the diagram

A B

C

ν

??~~~~~~~ν

__???????

can be completed to a commutative diagram after passing to K-theory:

K0(A) //_________ K0(B).

K0(C)

ν∗

::uuuuuuuuuν∗

ddIIIIIIIII

It would then be completely natural to use K-homology instead of K-theoryand ask that the first diagram can be completed to a commutative diagramafter passing to K-homology

K0(A)

ν∗

$$IIIII

IIIIK0(B).oo_ _ _ _ _ _ _ _ _

ν∗

zzuuuuuuuuu

K0(C)

Now let us observe that the condition, imposed in Proposition 3.1, thatν∗ : K0(D) → K0(C) is surjective clearly is equivalent to the existence of a



commutative diagram

K0(O∞)

ν∗

%%JJJJJJJJJ

K0(D)αoo

ν∗

zzuuuuuuuuu

K0(C)

where α is a surjective morphism.If D satisfies the UCT, then the condition above can be translated in terms

of K-theory as follows. Since the commutative diagram

K0(D)

ν∗

// Hom(K0(D),Z)

K0(C) // Hom(K0(C),Z)

has surjective horizontal arrows, the assumption on K-homology in Proposi-tion 3.1 is equivalent for the existence a group homomorphism K0(D) → Zwhich maps [1D] to 1. This is obviously equivalent to the condition that [1D]is an infinite order element of K0(D) and that the subgroup that it generates,Z[1D], is a direct summand of K0(D).

Our next goal is to show that for a unital Kirchberg algebra the propertyof being strongly self-absorbing is purely a KK -theoretical condition. Let

Cν = f : [0, 1]→ D | f(0) ∈ C1D, f(1) = 0be the mapping cone of the unital ∗-homomorphism ν : C→ D.

Proposition 3.3. Let D be a unital Kirchberg algebra. Then D is stronglyself-absorbing if and only if Cν ⊗D is KK-equivalent to zero.

Proof. We begin with a general observation. For a ∗-homomorphism ϕ : A→ Bof separable C∗-algebras and any separable C∗-algebra C, there is an exactPuppe sequence in KK -theory ([1, Thm. 19.4.3]):

KK (B,C)ϕ∗

// KK (A,C) // KK (Cϕ, C)

KK 1(Cϕ, C)

OO

KK 1(A,C)oo KK 1(B,C).ϕ∗

oo

It is apparent that [ϕ] ∈ KK (A,B)−1 if and only if composition with [ϕ] ∈KK (A,B) induces a bijection ϕ∗ : KK (B,C) → KK (A,C) for any separableC∗-algebra C, or equivalently, for just C = A and C = B. Therefore, by theexactness of the Puppe sequence, we see that that ϕ induces a KK -equivalenceif and only if its mapping cone C∗-algebra Cϕ is KK -contractible.

By applying this observation to the unital ∗-homomorphism ν ⊗ idD : D →D ⊗ D we deduce that ν ⊗ idD induces a KK -equivalence if and only if its



mapping cone Cν⊗idD∼= Cν ⊗D is KK -contractible. Suppose now that D is

a strongly self-absorbing Kirchberg algebra. Then ν ⊗ idD is asymptoticallyunitarily equivalent to a an isomorphism by [5, Thm. 2.2] and hence ν ⊗ idDinduces a KK -equivalence. Conversely, if ν ⊗ idD induces a KK -equivalence,then ν ⊗ idD is asymptotically unitarily equivalent to an isomorphism D →D ⊗D by [10, Thm. 8.3.3] and hence D is strongly self-absorbing.

We have the following result related to Proposition 3.3.

Proposition 3.4. Let D be a unital Kirchberg algebra such that D ∼= D ⊗D.The following assertions are equivalent:

(1) D is strongly self-absorbing.(2) KK (Cν , SD) = 0.(3) KK (Cν , D ⊗A) = 0 for all separable C∗-algebras A.(4) The map KK (D,D⊗A)→ KK (C, D⊗A) is bijective for all separable

C∗-algebras A.

Proof. (iii) ⇔ (iv). This equivalence is verified by using the Puppe sequenceassociated to ν : C→ D, arguing as in the proof of Proposition 3.3.

(i) ⇒ (iv). This implication is proved in [5, Thm. 3.4].(iii) ⇒ (ii). This follows by taking A = SC in (iii).(ii) ⇒ (i). Fix an isomorphism γ : D → D⊗D. Since KK 1(Cν , D⊗D) = 0

by hypothesis, it follows from the Puppe sequence that the map ν∗ : KK (D,D⊗D) → KK (C, D ⊗ D) is injective. Therefore γ and ν ⊗ idD induce the sameclass in KK (D,D ⊗D) since they are both unital. It follows that ν ⊗ idD isasymptotically unitarily equivalent to γ and soD is strongly self-absorbing.

Corollary 3.5. Let D be a unital Kirchberg algebra such that D ∼= D ⊗ D.Then D is strongly self-absorbing if and only if π2Aut(D) = 0.

Proof. Since π2Aut(D) ∼= KK (Cν , SD) by [5, Cor. 3.1], the conclusion followsfrom Proposition 3.4.

It was shown in [4, Prop. 4.1] that if a unital Kirchberg algebra satisfies theUCT, then D is strongly self-absorbing if and only if the homotopy classes[X,Aut(D)] reduces to a singleton for any path connected compact metrizablespace X .

References

[1] B. Blackadar, K-theory for operator algebras, Second edition, Cambridge Univ. Press,Cambridge, 1998. MR1656031 (99g:46104)

[2] B. Blackadar, A. Kumjian and M. Rørdam, Approximately central matrix units andthe structure of noncommutative tori, K-Theory 6 (1992), no. 3, 267–284. MR1189278(93i:46129)

[3] M. Dadarlat, The homotopy groups of the automorphism group of Kirchberg algebras,J. Noncommut. Geom. 1 (2007), no. 1, 113–139. MR2294191 (2008k:46157)

[4] M. Dadarlat, Fiberwise KK -equivalence of continuous fields of C∗-algebras, J. K-Theory3 (2009), no. 2, 205–219. MR2496447



[5] M. Dadarlat and W. Winter, On the KK -theory of strongly self-absorbing C∗-algebras,Math. Scand. 104 (2009), no. 1, 95–107. MR2498373

[6] G. Gong, X. Jiang and H. Su, Obstructions to Z-stability for unital simple C∗-algebras,Canad. Math. Bull. 43 (2000), no. 4, 418–426. MR1793944 (2001k:46086)

[7] U. Haagerup, Every quasi-trace on an exact C∗-algebra is a trace. Preprint 1991.[8] M. Rørdam, The stable and the real rank of Z-absorbing C∗-algebras, Internat. J. Math.

15 (2004), no. 10, 1065–1084. MR2106263 (2005k:46164)[9] M. Rørdam, and W. Winter, The Jiang-Su algebra revisited. To appear in J. Reine

Angew. Math.[10] M. Rørdam, Classification of nuclear, simple C∗-algebras, in Classification of nu-

clear C∗-algebras. Entropy in operator algebras, 1–145, Encyclopaedia Math. Sci., 126,Springer, Berlin. MR1878882 (2003i:46060)

[11] A. S. Toms and W. Winter, Strongly self-absorbing C∗-algebras, Trans. Amer. Math.Soc. 359 (2007), no. 8, 3999–4029 (electronic). MR2302521 (2008c:46086)

[12] A. S. Toms and W. Winter, Z-stable ASH algebras, Canad. J. Math. 60 (2008), no. 3,703–720. MR2414961

[13] W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C∗-algebras.Preprint.

Received February 23, 2009; accepted April 10, 2009

Marius DadarlatDepartment of Mathematics, Purdue UniversityWest Lafayette, IN 47906, USAE-mail: [email protected]

URL: http://www.math.purdue.edu/~mdd

Mikael RørdamDepartment of Mathematics, University of CopenhagenUniversitetsparken 5, 2100 Copenhagen Ø, DenmarkE-mail: [email protected]

URL: http://www.math.ku.dk/~rordam




Mahler measures and Fuglede-Kadison

determinants

Christopher Deninger

(Communicated by Linus Kramer)


Abstract. The Mahler measure of a function on the real d-torus is its geometric meanover the torus. It appears in number theory, ergodic theory and other fields. The Fuglede–Kadison determinant is defined in the context of von Neumann algebra theory and can beseen as a noncommutative generalization of the Mahler measure. In the paper we discuss andcompare theorems in both fields, especially approximation theorems by finite dimensional

determinants. We also explain how to view Fuglede–Kadison determinants as continuousfunctions on the space of marked groups.

1. Introduction

For an essentially bounded complex valued measurable function P on thereal d-torus T d = S1 × . . .× S1 the Mahler measure is defined by the formulaM(P ) = expm(P ) ≥ 0 where m(P ) is the integral

m(P ) =

∫

Tdlog |P | dµ in R ∪ −∞.

Here µ is the Haar probability measure on T d. If P is a Laurent polynomialon T d for example, it is known that log |P | is integrable on T d unless P = 0,so that we have M(P ) > 0 for P 6= 0 and M(P ) = 0 for P = 0.

The Mahler measure appears in many branches of mathematics. It is es-pecially interesting for polynomials with coefficients in Z. If α is an algebraicinteger with monic minimal polynomial P over Q then m(P ) is the normalizedWeil height of α. This follows from an application of Jensen’s formula. Forpolynomials in several variables there is no closed formula evaluatingm(P ) butsometimes m(P ) can be expressed in terms of special values of L-functions, seee.g. [2], [14] and their references.

The logarithmic Mahler measure m(P ) of a Laurent polynomial P over Zalso appears in ergodic theory as the entropy of a certain subshift defined by P

46 Christopher Deninger

of the full shift for Zd with values in the circle cp. [22] and [26]. For relationsof m(P ) with hyperbolic volumes we refer to [4].

We now turn our attention to the determinants in the title.Let N be a finite von Neumann algebra with a faithful normal finite trace

τ . In this note we only need the von Neumann algebra NΓ of a discrete groupΓ which is easy to define, cp. Section 2. For an invertible operator A in N theFuglede-Kadison determinant [12] is defined by the formula

detNA = exp τ(log |A|).

Here |A| = (A∗A)1/2 and log |A| are operators in N obtained by the functionalcalculus. For arbitrary operators A in N one sets

detNA = limε→0+

detN (|A|+ ε).

The main result in [12] asserts that detN is multiplicative on N . This deter-minant has several interesting applications. It appears in the definitions ofanalytic and combinatorial L2-torsion of Laplacians on covering spaces [19],[25] and [7], [21]. It was used in the work [13] on the invariant subspace prob-lem in II1-factors and it is related to the entropy of algebraic actions of discreteamenable groups [9], [10] and to Lyapunov exponents [8].

It was observed in [24, Example 3.13] that the Mahler measure has thefollowing functional analytic interpretation. For the group Γ = Zd there is acanonical isomorphism of NΓ with L∞(T d, µ) which we write as A 7→ A. Therelation of Mahler measures with Fuglede-Kadison determinants is then givenby the formula:

(1) detNZd(A) = M(A) for all A ∈ NZd.

In this note we review certain classical properties of Mahler measures anddiscuss their generalizations to Fuglede-Kadison determinants of group vonNeumann algebras. In particular, this concerns approximation formulas e.g. byfinite dimensional determinants. Usually the results for Mahler measures arestronger than the corresponding ones for general Fuglede-Kadison determi-nants and this raises interesting questions. In Section 2 we also extend partof the formalism of the theory of orthogonal polynomials on the unit circle toa noncommutative context. Moreover, in Section 3 we show that in a suitablesense detNΓ is continuous on the space of marked groups if the argument isinvertible in L1.

2. Approximation by finite dimensional determinants

In this section we discuss one way to approximate Mahler measures andmore generally Fuglede-Kadison determinants of amenable groups by finitedimensional determinants. Another method which works for residually finitegroups is explained in the next section as a special case of Theorem 17.


Mahler measures and Fuglede-Kadison determinants 47

The Mahler measure aspect of this topic begins with Szego’s paper [29]. Foran integrable function P on S1 consider the Fourier coefficients

cν =

∫

S1

z−νP (z) dµ(z) for ν ∈ Z

and define the following determinants for n ≥ 0

Dn = det

c0 c1 cn−1

c−1 c0 cn−2

c−n+1 c−n+2 c0

.

If P is real valued we have cν = c−ν and if P (z) ≥ 0 for all z ∈ S1 we mayview c−ν as the ν-th moment of the measure P (z) dµ(z). In that case the Dn’sare the associated Toeplitz determinants.

Theorem 1 (Szego). If P is a continuous real valued function on S1 withP (z) > 0 for all z ∈ S1, then Dn > 0 for all n ≥ 0 and we have the limitformula:

M(P ) = limn→∞

n√Dn.

Using the theory of orthogonal polynomials on the unit circle the conditionsin Szego’s original theorem have been significantly relaxed, see [28] for thehistory:

Theorem 2. The assertions in Szego’s theorem hold for every real-valued non-negative essentially bounded measurable function P on S1 which is nonzero ona set of positive measure.

Proof. The Dn’s are determinants of Toeplitz matrices for the nontrivial mea-sure P dµ. These matrices are positive definite and in particular Dn > 0 forevery n ≥ 0, cp. [28, Section 1.3.2]. The limit formula M(P ) = limn→∞

n√Dn

is a special case of [28, Theorem 2.7.14], equality of (i) with (vi) applied to theprobability measure P‖P‖−1

1 dµ on S1. (In following that proof, the shortcutin the remark on p. 139 of loc. cit. is useful.)

Let us now explain the von Neumann aspect of these results. For a discretegroup Γ we will view the elements of Lp(Γ) as formal series

∑γ∈Γ xγγ with∑

γ |xγ |p < ∞. It is then clear that Γ acts isometrically by left and right

multiplication on Lp(Γ). The von Neumann algebraNΓ of Γ may be defined asthe algebra of bounded operators A : L2Γ→ L2Γ which are left Γ-equivariant.For γ ∈ Γ define the unitary operator Rγ : L2Γ → L2Γ by Rγ(x) = xγ. TheC-algebra homorphism

r : CΓ→ NΓ with r(∑γfγγ

)=∑γfγRγ−1

extends to a homomorphism r : L1(Γ) → NΓ with ‖r(f)‖ ≤ ‖f‖1 for allf ∈ L1Γ. By looking at r(f)(e) where e ∈ Γ ⊂ L2Γ is the unit element of Γ,we see that r is injective. It will often be viewed as an inclusion in the following.



Setting f∗ =∑fγγ

−1 for f =∑fγγ in L1Γ the equality r(f∗) = r(f)∗ holds.

The canonical trace τ = τNΓ on NΓ is defined by the formula τ(A) = (Ae, e).It vanishes on commutators [A,B] = AB −BA for A,B in NΓ. For f in L1Γwe have τ(r(f)) = fe. Finally, detNΓA is defined as in the introduction forevery A in NΓ.

For an abelian group Γ with (compact) Pontryagin dual Γ = Homcont(Γ, S1)

and Haar probability measure µ on Γ, the Fourier transform provides an isom-etry of Hilbert spaces

F : L2Γ∼−→ L2(Γ, µ).

On the dense subspace CΓ it is given by F(f)(χ) =∑γ fγχ(γ) for χ ∈ Γ.

One can show that under the induced isomorphism of algebras of boundedoperators

B(L2(Γ))→ B(L2(Γ, µ)), A 7→ F A F−1

the von Neumann algebra NΓ maps isomorphically onto L∞(Γ, µ) where the

latter operates by multiplication on L2(Γ, µ). Denoting this isomorphism by

A 7→ A we have A = F(A(e)). Namely, L2(Γ) is a left CΓ-algebra and forf ∈ CΓ we therefore have

F(A(f)) = F(fA(e)) = F(f)F(A(e)).

Now the assertion follows because F(CΓ) is dense in L2(Γ, µ). It follows thatwe have

τ(A) = (Ae, e) = (F(A(e)),F(e)) = (A, 1) =

∫

Γ

A dµ.

Hence there is a commutative diagram

NΓ∼ //

τ A

AAAA

AAA L∞(Γ, µ)

∫Γwwwwwwwww

C

where∫Γ

denotes integration against the measure µ. We conclude using thedefinition of detNΓ and Levi’s theorem that we have:

detNΓA = exp

∫

Γ

log |A| dµ for A ∈ NΓ.

In particular, for Γ = Zd we get formula (1) from the introduction. It alsofollows that the generalized Mahler measures studied in [17] can be expressedas Fuglede-Kadison determinants.

The noncommutative generalization of Szego’s theorem that we have inmind is valid for amenable groups. A Følner sequence (Fn) in Γ is a sequenceof finite subsets Fn ⊂ Γ which are almost invariant in the following sense: Forany γ ∈ Γ we have

limn→∞

|Fnγ \ Fn||Fn|

= 0.



A countable discrete group Γ is said to be amenable if it has a Følner sequence.For example Z is amenable, the sets Fn = 0, 1, . . . , n − 1 forming a Følnersequence.

For a finite subset F ⊂ Γ and an operator A ∈ NΓ consider the followingendomorphism of CF , the finite-dimensional C-vector space over F :

AF : CFiF→ L2Γ

A−→ L2ΓpF−−→ CF.

Here iF is the inclusion and pF the orthogonal projection to CF . We havep∗F = iF for the L2-adjoints and hence (AF )∗ = (A∗)F .

Lemma 3. If A ∈ NΓ is positive then AF is positive as well and hencedetAF ≥ 0. If A is positive, and injective on CΓ then AF is a positive auto-morphism of CF and hence detAF > 0.

Proof. Set B =√A. For v ∈ CF we have (AF v, v) = (Av, v) = ‖Bv‖2

and hence AF is positive. Moreover AF v = 0 implies Bv = 0 and henceAv = B(Bv) = 0. If A is injective on CΓ we get v = 0 and thus AF is injectiveand hence an automorphism.

The approximation result corresponding to Szego’s theorem is the followingone which was proved in [9, Theorem 3.2]:

Theorem 4. Let Γ be a finitely generated amenable group with a Følner se-quence (Fn) and let A be a positive invertible operator in NΓ. Then detAFn >0 for all n and we have:

detNΓA = limn→∞

(detAFn)1/|Fn|.

Positivity of detAFn follows from Lemma 3. The proof of the limit formulais based on an approximation result for traces of polynomials in A due toSchick [27] generalizing previous work of Luck. Theorem 4 follows by applyingthe Weierstraß approximation theorem to log and the fact that the spectrumof A and all AFn is uniformly bounded away from zero.

We would like to point out that another part of Szego’s theory which char-acterizes M(P ) by an extremal property has been generalized to the setting ofvon Neumann algebras in [1].

Example 5. Let us now show that Szego’s Theorem 1 is a special case of The-orem 4. Consider a measurable essentially bounded function P : S1 → R withP (z) ≥ 0 for all z ∈ S1. It defines a positive element P of the von Neumann

algebra L∞(S1, µ). Let A be the positive operator in NZ with A = P , i.e.with F(A(0)) = P . For ν ∈ Z ⊂ L2(Z) write (ν) for its image in L2(Z). Thenwe have F(ν) = zν viewed as a character on S1. Thus

cν =

∫

S1

z−νP (z) dµ(z) = (P, zν) = (F(A(0)),F(ν)) = (A(0), (ν)).

Now consider the Følner sequence Fn = 0, 1, . . . , n− 1 of Z. The matrix ofAFn with respect to the basis (0), (1), . . . , (n−1) of CFn has (i, j)-th coefficient

(AFn(i), (j)) = (A(i), (j)) = (A(0), (j − i)) = cj−i.



Thus we have detAFn = Dn and therefore Theorem 4 implies Theorem 1 (evenwith “continuous” replaced by “measurable essentially bounded”).

Note that using other Følner sequences for Z, Theorem 4 gives new limitformulas for the Mahler measure not covered by Szego’s theorem.

The analogue of Theorem 2 in our setting does not seem to be known. Weformulate it as a question:

Question 6. Let Γ be a finitely generated amenable group and A a positiveoperator in NΓ. Does the limit formula

detNΓA = limn→∞

(detAFn)1/|Fn|

hold for every Følner sequence?

Remarks 1) In Theorem 2 the nonzero positive operators in NZ ∼= L∞(S1, µ)were considered. These are injective on CZ because F(CZ) = C[z, z−1], andnonzero Laurent polynomials vanish only in a set of measure zero on S1. Per-haps it is reasonable therefore to first consider only positive operators whichare injective on CΓ so that by Lemma 3 all AFn are positive automorphisms.On the other hand, for A = 0 the limit formula is trivially true.2) Because of the next proposition it would suffice to prove the inequality

detNΓA ≤ limn→∞

(detAFn)1/|Fn|

in order to answer Question 5 affirmatively.

Proposition 7. For a finitely generated group Γ and any positive operator Aon NΓ we have

detNΓA ≥ limn→∞

(detAFn)1/|Fn|.

Proof. For A in ZΓ this is proved in [27]. In general we can argue as follows.For any endomorphism ϕ set ϕ(ε) = ϕ+ εid. Then we have (A(ε))F = (AF )(ε)

for finite F ⊂ Γ. The following relations hold:

detNΓA(i)= lim

ε→0+detNΓA

(ε) (ii)= lim

ε→0+limn→∞

(det(A(ε))Fn)1/|Fn|

(iii)

≥ limn→∞

(detAFn)1/|Fn|.

Here (i) is true by the definition of the Fuglede-Kadison determinant and (ii)follows from Theorem 4 applied to A(ε). Finally (iii) holds because det(AFn)(ε)

≥ detAFn for every n ≥ 1 and ε > 0.

In the rest of this section we develop a formalism for the determinants detAFand detNΓA which is suggested by the theory of orthogonal polynomials onthe unit circle. We also point out the relation to Question 6.

We start with the following well known lemma:

Lemma 8. For a block matrix over a field with A invertible the followingformula holds:

det (A BC D ) = det(D − CA−1B) detA.



Proof. We have

det (A BC D ) = det

(I B

CA−1 D

)det (A 0

0 I ) = det(I B0 D−CA−1B

)detA.

Consider a countable discrete group Γ and finite subsets F ⊂ F ′ ⊂ Γ. LetA ∈ NΓ be positive, and injective on CΓ, so that according to Lemma 3 theendomorphism AF is positive and invertible. In terms of the decomposition

CF ′ = CF ⊕ C(F ′ \ F )

the endomorphism AF ′ is given by the block matrix

AF ′ =

(AF pFAiF ′\F

pF ′\FAiF pF ′\FAiF ′\F

).

Thus Lemma 8 gives the formula

(2) detAF ′ = detAF det(pF ′\FAiF ′\F − pF ′\FAiFA−1F pFAiF ′\F ).

Now consider the endomorphism

ψ = iFA−1F pFAiF ′ : CF ′ → CF ′

and the scalar product on CΓ defined by

(u, v)A := (Au, v) = (u,Av).

It is positive since for u, v ∈ CΓ there is a finite subset F ⊂ Γ with u, v ∈ CFand then we have (u, v)A = (AFu, v) with the positive automorphism AF .

Proposition 9. The endomorphism ψ is the orthogonal projection of CF ′ toCF with respect to the scalar product (, )A on CF ′.

Proof. For u ∈ CF we have ψ(u) = A−1F AFu = u. This implies that ψ2 = ψ

since ψ takes values in CF . Moreover, Imψ = CF . Next observe that

pF ′Aψ = pF ′AiFA−1F pFAiF ′

is selfadjoint since i∗F = pF and A,AF are selfadjoint. Hence we have pF ′Aψ =ψ∗AiF ′ and for u, v ∈ CF ′ therefore:

(ψu, v)A = (ψu,Av) = (u, ψ∗Av) = (u,Aψv) = (u, ψv)A.

By the proposition the endomorphism ϕ = id− ψ of CF ′ is the orthogonalprojection to CF⊥A with respect to (, )A. Formula (2) can be rewritten as

detAF ′ = detAF det(pF ′\FAϕiF ′\F ).

Corollary 10. Assume that F ′ = F ∪γ and set Φγ = ϕ(γ). Then we havedetAF ′ = ‖Φγ‖2A detAF .



Proof. Since C(F ′ \ F ) = Cγ is one-dimensional and ‖γ‖ = 1, we have

det(pF ′\FAϕiF ′\F ) = (pF ′\FAϕ(γ), γ) = (Aϕ(γ), γ) = (ϕ(γ), γ)A

= (ϕ2(γ), γ)A = (ϕ(γ), ϕ(γ))A = ‖ϕ(γ)‖2A.

The corollary generalizes part of formula (1.5.78) of [28]. Using this orthog-onalization process inductively we get the formula

detAF =∏

γ∈F

‖Φγ‖2A.

Concerning the order of ‖Φγ‖A note the following equations

‖Φγ‖2A = (ϕ(γ), γ)A = (γ − iFA−1F pFAγ, γ)A

= ‖γ‖2A − (iFA−1F pFAγ,Aγ)

= τ(A) − (A−1F s, s), where s = pFAγ ∈ CF.

In the situation of Corollary 10 we therefore obtain:

Corollary 11. We have 0 < ‖Φγ‖2A ≤ τ(A). Moreover the following asser-tions are equivalent:1) ‖Φγ‖2A = τ(A)2) pFAγ = 03) AF ′ =

(AF 00 c

)for some c (which must be c = τ(A)).

Now we generalize a calculation from the theory of orthogonal polynomialson S1 which is used in one of the proofs of Theorem 2. Recall that for Φ ∈ NΓone sets ‖Φ‖2 = τ(Φ∗Φ)1/2. It is known that we have

(3) detNΓΦ ≤ ‖Φ‖2.Namely, let Eλ be the spectral resolution of |Φ|. Then we have by Jensen’sinequality:

(detNΓΦ)2 = exp

∫ ∞

0

log(|λ|2)dτ(Eλ) ≤∫ ∞

0

|λ|2 dτ(Eλ)

= τ( ∫ ∞

0

|λ|2dEλ)

= τ(Φ∗Φ) = ‖Φ‖22.

For positive A ∈ NΓ and any Φ ∈ NΓ we find

(detNΓA)1/2detNΓΦ = detNΓ(√AΦ) ≤ ‖

√AΦ‖2

= τ(Φ∗AΦ)1/2 = (Φ∗AΦe, e)1/2

= (AΦ(e),Φ(e))1/2 = ‖Φ(e)‖A.Let ∼: CΓ → CΓ be defined by f =

∑fγγ−1. Then for f ∈ CΓ the operator

r(f) ∈ NΓ is right multiplication by f . For f ∈ CΓ ⊂ NΓ where the inclusionis via r, we get

(detNΓA)1/2detNΓf ≤ ‖f(e)‖A = ‖f‖A.



Applying this to f = Φγ we find

(detNΓA)1/2detNΓΦγ ≤ ‖Φγ‖A.Combining this with Corollary 10 we obtain the following result:

Corollary 12. Let A ∈ NΓ be positive, and injective on CΓ. Assume that Fand F ′ = F ∪γ are finite subsets of Γ. Then we have the inequality:

(detNΓA)(detNΓΦγ)2 ≤ detAF ′

detAF.

Remark 13. Consider a function P : S1 → R as in Example 5 and assumethat

∫S1 P dµ = 1. Then µP = P dµ is a probability measure on S1 and we may

consider the orthogonal projection ϕn of 〈1, z, . . . , zn〉 onto 〈1, z, . . . , zn−1〉⊥ inL2(S1, µP ). The monic polynomial of degree n given by Φn(z) = ϕn[zn] is then-th orthogonal polynomial with respect to µP . It is known that all zeroes ofΦn(z) lie in the open unit disc. Following [28, p. 102], the shortest argumentfor this fundamental fact seems to be the following. We write ‖ ‖P for thenorm corresponding to the scalar product (α, β)P := (Pα, β)2 = (α, Pβ)2 ofL2(S1, µP ). Let z0 ∈ C be a zero of Φn(z). Then we have zf = Φn + z0f fora polynomial f of degree n− 1. This gives the equation

‖f‖2P = ‖zf‖2P = ‖Φn‖2P + |z0|2‖f‖2P ,since f ∈ 〈1, z, . . . , zn−1〉 implies (Φn, f)P = 0. It follows that |z0| < 1.

As a consequence, note that Jensen’s formula implies that M(Φn(z)) = 1.For Γ = Z and F = 0, . . . , n − 1 and F ′ = 0, . . . , n we have γ = n in

our above notation. Consider the operator A ∈ NΓ with A = P i.e. withF(A(0)) = P . For Φγ defined as in Corollary 10, one checks that we haveF(Φγ) = Φn(z).

Hence we get

detNZΦγ = M(F(Φγ)) = M(Φn(z)) = 1.

Thus in this special case the inequality in Corollary 12 gives

detNΓA ≤detAFn

detAFn−1

where Fn = 0, . . . , n. This inequality is instrumental for the proof of Theo-

rem 2. For general Γ, unfortunately we do not know whether detNΓ Φγ ≥ 1 or

even detNΓ Φγ = 1 holds under suitable conditions.

3. Approximation on the space of marked groups

According to a theorem of Lawton, Mahler measures of Laurent polynomialsin several variables can be approximated by Mahler measures of one-variableLaurent polynomials. His result which we now recall resolved a conjecture ofBoyd. For r ∈ Zd set

q(r) = min‖ν‖ | 0 6= ν ∈ Zd with (ν, r) = 0where ‖ν‖ = max |νi| and (ν, r) =

∑i νiri.



Theorem 14 (Lawton [15]). For r ∈ Zd and P in C[X±11 , . . . , X±1

d ] setPr(X) = P (Xr1 , . . . , Xrd). Then we have

limq(r)→∞

M(Pr) = M(P ).

If P does not vanish on T d, so that log |Pr| is continuous on S1 the theoremis much simpler to prove than in general. In the following we will generalize thiseasy case to a statement on the continuity of the Fuglede-Kadison determinanton the space of marked groups. A full generalization of Theorem 14 in thisdirection is a challenging problem.

For d ≥ 1 the space Xd of marked groups on d-generators is the set ofisomorphism classes [Γ, S] of pairs (Γ, S) where Γ is a discrete group and S =(s1, . . . , sd) a family of d generators of Γ. Here repetitions are allowed. Twosuch pairs (Γ, S) and (Γ′, S′) are called isomorphic if there is an isomorphism

α : Γ∼−→ Γ′ with α(S) = S′. The set Xd becomes an ultra-metric space with

the distance function

d([Γ1, S1], [Γ2, S2]) = 2−N

where N ≤ ∞ is the largest radius such that the balls of radius N around theorigin in the Cayley graphs of (Γ1, S1) and (Γ2, S2) are isomorphic as oriented,labelled graphs with labels 1, . . . , d corresponding to the generators. Thusintuitively two marked groups are close to each other if their Cayley graphsaround the origin coincide on a big ball. An equivalent metric onXd is obtainedby setting

δ([Γ1, S1], [Γ2, S2]) = 2−M

if the bijection S1∼= S2 induces a bijection of S1- resp. S2-relations of length

less than M and if M ≤ ∞ is maximal with this property. Here an S-relationin a group Γ is an S-word, i.e. a finite string of elements from S and theirinverses, whose evalution in Γ is equal to e. The number of elements in thestring defining a word is the length of the word e.g. s−1

1 s2s1s−13 s5 has length

5.Much more background on the space of marked groups can be found in [6,

Section 2], for example.

Example 15. With notations as in Theorem 14 consider

(Γ, S) = (Zd, e1, . . . , ed) and (Γr, Sr) = (D(r)Z, r1 , . . . , rd)

where r ∈ Zd and D(r) is the greatest common divisor of r1, . . . , rd. Then wehave

limq(r)→∞

[Γr, Sr] = [Γ, S] in Xd.

Proof. As Γr is abelian, an Sr-word is a relation in Γr if and only if∑d

i=1 νiri =0 where νi ∈ Z is the sum of all exponents ±1 of ri in the word. The lengthof the relation is at least ‖ν‖. If a relation R has length less than q(r) itfollows that we have ν = 0 and hence R is a relation of commutation. Hence



for length less than q(r) the relations in (Γ, S) and (Γr, Sr) are in canonicalbijection. Thus we have

δ([Γ, S], [Γr, Sr]) ≤ 2−q(r)

and the assertion follows.

Example 16. Let Γ be a countable group and (Kn) a sequence of normalsubgroups of Γ. We write Kn → e if e is the only element of Γ which is con-tained in infinitely many Kn’s. Equivalently, for any finite subset Q ⊂ Γ wehave Kn ∩Q ⊂ e for n large enough.Now assume that Γ is finitely generated and let S be a finite family of genera-tors. Given epimorphisms ϕn : Γ ։ Γn we get finite families of generators Snin Γn. Setting d = |S|, we claim that the limit formula

limn→∞

[Γn, Sn] = [Γ, S] in Xd

is equivalent to Kn → e, where Kn = Kerϕn.

Proof. Assume that Kn → e. Let Rn be a relation of length l in Γn and let Rbe the corresponding S-word in Γ. The evaluation γ = ev(R) of R in Γ lies inKn. Let Q ⊂ Γ be the finite subset of at most l-fold products from S ∪ S−1.In particular γ ∈ Q. For n ≥ n(l), we have Kn ∩ Q ⊂ e since Kn → e.Therefore the relations of length ≤ l in Γ and Γn are in canonical bijection ifn ≥ n(l) and hence we have

δ([Γn, Sn], [Γ, S]) ≤ 2−l for n ≥ n(l).

For the converse consider an element γ ∈ Γ which is contained in infinitelymany Kn’s. Choose a word W in Γ with γ = ev(W) and let l be the lengthof W . By assumption, there are arbitrarily large n’s such that ϕn(W) is arelation in Γn. But for n ≫ 0 the relations of length l in Γn and Γ are inbijection. Hence W must be a relation i.e. γ = ev(W) = e.

In order to state the next result we introduce some notations.For a homomorphism ϕ : Γ→ Γ′ of discrete groups denote by ϕ∗ : L1(Γ)→

L1(Γ′) the map “integration along the fibres” defined by

ϕ∗

( ∑γ∈Γ

fγγ)

=∑γ∈Γ

fγϕ(γ) =∑γ′∈Γ′

( ∑γ∈ϕ−1(γ′)

fγ

)γ′.

The map ϕ∗ is a homomorphism of Banach ∗-algebras with units and it satisfiesthe estimate ‖ϕ∗(f)‖1 ≤ ‖f‖1 for all f ∈ L1(Γ).

Recall that we view L1(Γ) as a subalgebra of NΓ. Let L1(Γ)× be the groupof invertible elements in L1(Γ). Then we have the following result

Theorem 17. Consider a countable discrete group together with homomor-phisms ϕn : Γ → Γn. For f ∈ L1(Γ) set fn = ϕn∗(f) ∈ L1(Γn). Then wehave:

(4) detNΓf ≥ limn→∞

detNΓnfn if Kn = Kerϕn → e.



In case f ∈ L1(Γ)×, equality holds:

(5) detNΓf = limn→∞

detNΓnfn if Kn → e.

In particular, using Example 16 we get the following corollary:

Corollary 18. For [Γ, S] ∈ Xd, epimorphisms ϕn : Γ ։ Γn and f ∈ L1(Γ)×

we have

detNΓf = limn→∞

detNΓnfn if [Γn, Sn]→ [Γ, S] in Xd.

Let us give two examples:

Example 19. For any countable residually finite group Γ there is a sequenceof normal subgroups Kn with finite index such that Kn → e. Set Γn = Kn \Γ.Then we have:


| det r(fn)|1/|Γn| for any f ∈ L1(Γ)×.

Note here that r(fn) ∈ NΓn ⊂ End CΓn. This formula follows immediatelyfrom Theorem 17 if we note that for a finite group G and an element h ∈ CG =L1(G) we have:

detNGh = | det r(h)|1/|G|.Formula (6) was used in [10] to relate the growth rate of periodic points ofcertain algebraic Γ-actions to Fuglede-Kadison determinants. For f in ZΓ ∩L1(Γ)× formula (6) is a special case of [23, Theorem 3.4, 3].

Example 20. Recall the situation of Example 15 and let P be a continuousfunction on T d whose Fourier coefficients are absolutely summable. Thus wehave P = F(f) for some f ∈ L1(Zd). If we assume that P does not vanishin any point of T d it follows from a theorem of Wiener [30] that we havef ∈ L1(Zd)×. Define ϕr : Γ = Zd → Γr by ϕr(ei) = ri for 1 ≤ i ≤ r. Corollary18 now implies the formula

detNΓf = limq(r)→∞

detNΓrfr.

Since detNΓ f = M(P ) and detNΓrfr = M(Pr), we get the limit formula ofLawton’s theorem in this (easy) case.

The theorem of Wiener mentioned above has been generalized to the non-commutative context. The ultimate result is due to Losert [18]. It asserts thatL1(Γ)× = L1(Γ)∩C∗(Γ)× if and only if Γ is “symmetric”. Thus for symmetricgroups the question of invertibility in L1(Γ) is reduced to the easier questionof invertibility in the C∗-algebra C∗(Γ). Finitely generated virtually nilpotentdiscrete groups for example are known to be symmetric [20, Corollary 3], andhence we have the following equalities for them

(7) L1(Γ)× = L1(Γ) ∩ C∗r (Γ)× = L1(Γ) ∩ (NΓ)×.

Note that for amenable groups the C∗-algebra and the reduced C∗-algebracoincide. The classical Wiener theorem is a special case of (7):

L1(Zd)× = L1(Zd) ∩ C0(T d)× = L1(Zd) ∩ L∞(T d, µ)×.



The assumptions in Corollary 18 are more restrictive than in Theorem 17. Theadvantage of its formulation lies in the intuition and results about Xd that onemay use.Proof of Theorem 17 First we need a simple result about traces. We claimthat for any f in L1(Γ) and any complex polynomial P (X) we have

(8) τNΓ(P (f)) = limn→∞

τNΓn(P (fn)) if Kn → e.

Since P (f) lies in L1(Γ) as well, it suffices to prove (8) for P (X) = X . Writingf =

∑γ fγγ we have τNΓ(f) = fe and τNΓn(fn) =

∑γ∈Kn

fγ . Fix ε > 0. Since

f is in L1(Γ), there is a finite subset Q ⊂ Γ with∑γ∈Γ\Q |fγ | < ε. Because of

the assumption Kn → e, there is some N ≥ 1 such that Kn ∩Q ⊂ e for alln ≥ N . For n ≥ N we therefore get the estimate

|τNΓ(f)− τNΓn(fn)| = |fe −∑

γ∈Kn

fγ | ≤∑

γ∈Kn\e

|fγ | ≤∑

γ∈Γ\Q

|fγ | < ε.

Since ε > 0 was arbitrary, formula (8) follows.Next, for any f ∈ L1(Γ) we have

(9) ‖r(fn)‖ ≤ ‖fn‖1 ≤ ‖f‖1 and ‖r(f)‖ ≤ ‖f‖1where ‖ ‖ is the operator norm (between L2-spaces).

Moreover, if f ∈ L1(Γ)×, the relation (f−1)n = ϕn(f−1) = ϕn(f)−1 = f−1n

implies the estimates:

(10) ‖r(f−1n )‖ ≤ ‖f−1

n ‖1 ≤ ‖f−1‖1.Since 2 detNΓ f = detNΓ f

∗f and

(f∗f)n = ϕn(f∗f) = ϕn(f)∗ϕn(f) = f∗nfn

we may replace f by f∗f in the assertion of Theorem 17. Hence we mayassume that f ∈ L1(Γ) and fn ∈ L1(Γn) are positive in NΓ resp. NΓn i.e.that r(f) and r(fn) are positive operators. If f is invertible it follows that thespectrum of r(f) is contained in the interval I = [‖f−1‖−1

1 , ‖f‖1]. Using theestimates (9) and (10) we see that the spectra of r(fn) lie in I as well for all n.Note here that for a positive bounded operator A on a Hilbert space we have‖A‖ = maxλ∈σ(A) λ. Fix ε > 0. Since I is a compact subinterval of (0,∞), itfollows from the Weierstraß approximation theorem that there is a polynomialP (X) with maxx∈I |P (x)− log x| ≤ ε. Since σ(r(f)), σ(r(fn)) lie in I it followsthat we have:

‖ log r(f)− P (r(f))‖ ≤ ε and ‖ log r(fn)− P (r(fn))‖ ≤ ε.Using the estimate |τNΓA| = |(Ae, e)| ≤ ‖A‖ for any A ∈ NΓ, we obtain:

|τNΓ(log r(f))− τNΓn(log r(fn))| ≤ 2ε+ |τNΓ(P (f))− τNΓn(P (fn))|.Assertion (8) now implies formula (5) in Theorem 17. For the proof of (4)we can assume as above that r(f) and the r(fn) are positive operators. Therelations (9) imply that the spectra of r(f) and of all r(fn) lie in J = [0, ‖f‖1].Choose a sequence of polynomials Pk(X) ∈ R[X ]. converging pointwise to



log in J and satisfying the inequalities Pk > Pk+1 > log in J for all k. Onemay obtain such a sequence (Pk) as follows. The continuous functions ϕk onJ defined by ϕk(x) = 1/k + log x for x ≥ 1/k and by ϕk(x) = 1/k + log 1/kfor 0 ≤ x ≤ 1/k satisfy the inequalities ϕk > ϕk+1 > log in J and convergepointwise to log in J . Setting

ψk = 2−1(ϕk + ϕk+1) and εk = minx∈J

(ϕk(x) − ϕk+1(x)) > 0,

the Weierstraß approximation theorem provides us with polynomials Pk suchthat

maxx∈J|ψk(x)− Pk(x)| ≤

εk4.

They satisfy the estimates ϕk > Pk > ϕk+1 for all k and hence have the desiredproperties. It follows that we have

(11) limk→∞

τNΓ(Pk(f)) = log detNΓf.

To see this, consider the spectral resolution Eλ of the operator r(f). Then wehave by the definition of detNΓ f :

log detNΓf = limε→0+

τNΓ(log(r(f) + ε)) = limε→0+

∫

J

log(λ+ ε) dτNΓ(Eλ)

(a)=

∫

J

logλdτNΓ(Eλ)

(b)= lim

k→∞

∫

J

Pk(λ) dτNΓ(Eλ)

= limk→∞

τNΓ(Pk(f)).

Here equations (a) and (b) hold because of Levi’s theorem in integration theory(with respect to the finite measure dτNΓ(Eλ) on J). Noting the estimate

(12) τNΓn(Pk(fn)) ≥ τNΓn(log(fn))

we obtain the relations:

log detNΓf(11)= lim

k→∞τNΓ(Pk(f))

(8)= lim

k→∞limn→∞

τNΓn(Pk(fn))

(12)

≥ limn→∞

τNΓn(log fn) = limn→∞

log detNΓnfn.

2

Remark If f ∈ L1(Γ) is not invertible the question whether the equality(5) still holds becomes much more subtle. In the situation of Example 19,Luck has given a criterion in terms of the asymptotic behavior near zero of thespectral density function, which is hard to verify however, cp. [23, Theorem 3.4,3]. Note that he discusses a slightly different version of the Fuglede-Kadisondeterminant where the zero-eigenspace is discarded. If A ∈ NΓ is injective onL2(Γ) the two versions of the FK-determinant agree. Incidentally, for a finitelygenerated amenable group Γ, a nonzero divisor f ∈ CΓ has the property thatr(f) is injective on L2(Γ), see [11].



For Γ = Z and the projections to Γn = Z/n the above question is relatedto the theory of diophantine approximation. This was first noted in ergodictheory because for f ∈ Z[Z] the limit

limn→∞

log detNZ/n(fn) = limn→∞

n−1 log det(r(fn)) (if it exists)

is the logarithmic growth rate of the number of periodic points of a toral

automorphism with characteristic polynomial f ∈ Z[X±1]. One wanted toknow if it is equal to the topological entropy which turns out to be given by

m(f) = log detNZ(f). Using a theorem of Gelfond this was proved by Lind in[16, § 4]. See also [26, Lemma 13.53]. On the other hand there are examples

of noninvertible f ∈ L1(Z) with f ∈ R[X,X−1] a linear polynomial for whichformula (5) is false, see [24, Example 13.69].

On the other hand, for the sequence ϕr : Γ = Zd → Γr from Example 15formula (5) holds for all f ∈ C[Zd] as follows from Lawton’s Theorem 14 above.One may interpret his proof as an estimate for the spectral density function of|f | near zero.

These cases suggest the following problem:

Question 21. In the situation of Theorem 17 consider f in ZΓ. Is it truethat we have

detNΓf = limn→∞

detNΓnfn if Kn → e

even if f is not invertible in L1(Γ)?

In the rest of this section we extend the previous theory somewhat by re-placing the maps ϕn : Γ → Γn by a sequence of “correspondences”. Thus, weconsider discrete groups and homomorphisms

Γϕ←− Γ

ϕn−−→ Γn with kernels K = Kerϕ and Kn = Kerϕn.

Given f ∈ L1(Γ) write f = ϕ∗(f) ∈ L1(Γ) and fn = ϕn∗(f) ∈ L1(Γn). We willwrite Kn → K if one of the following equivalent conditions holds:

a No element γ ∈ Γ is contained in K Kn for infinitely many n.b For any finite subset Q ⊂ Γ we have (KKn)∩Q = ∅ if n is large enough.

Then Theorem 17 has the following generalization:

Theorem 22. Consider diagrams of countable groups Γϕ←− Γ

ϕn−−→ Γn forn ≥ 1 as above and fix f ∈ L1(Γ). Then we have

(13) detNΓf ≥ limn→∞

detNΓnfn if Kn → K.

For f ∈ L1(Γ)×, equality holds:


detNΓnfn if Kn → K.

Proof. As before one first shows that:

(15) τNΓ(P (f)) = limn→∞

τNΓn(P (fn)) for Kn → K



whenever f ∈ L1(Γ) and P ∈ C[X ]. Writing f =∑

γ∈Γ aγ γ and using theinequality

|τNΓ(f)− τNΓn(fn)| ≤∑

γ∈KKn

|aγ |,

we can argue as in the proof of formula (8).The rest of the proof is analogous to the one of Theorem 17 if we note

that the spectra of r(f) and r(fn) lie in [0, ‖f‖1] for f ∈ L1(Γ) and in

[‖f−1‖−11 , ‖f‖1] if f is invertible in L1(Γ). This follows from the estimates

‖r(f)‖ ≤ ‖f‖1 ≤ ‖f‖1 and ‖r(fn)‖ ≤ ‖fn‖1 ≤ ‖f‖1 if f ∈ L1(Γ)

and similar ones for the inverses of f , f, fn in case f ∈ L1(Γ)×.

Next, assume that Γ is finitely generated and the maps ϕ and ϕn are sur-jective. A family of generators S of Γ gives families of generators S and Snfor Γ and Γn. If d = |S| one can show as in Example 16 that the conditionKn → K is equivalent to [Γn, Sn]→ [Γ, S] in Xd for n→∞.For completeness let us give the argument for the implication needed in the fol-lowing corollary. Assume that γ ∈ Γ is contained in KKn for infinitely manyn. Choose a word W in Γ with γ = ev(W). Via ϕn, ϕ we obtain words Wn

and W in Γn resp. Γ with γn = ev(Wn) and γ = ev(W). By assumption thereare infinitely many n, such that W is a relation in Γ but Wn is not a relationin Γn or vice versa. This is not possible however, since for large n the relationsof length ≤ l(W) in Γ and Γn are in canonical bijection if [Γn, Sn]→ [Γ, S].

Corollary 23. a In the situation above, we have for f ∈ L1(Γ)

limn→∞

τNΓn(fn) = τNΓ(f) if [Γn, Sn]→ [Γ, S] in Xd.

b If f is invertible in L1(Γ), we have in addition

limn→∞

detNΓn(fn) = detNΓ(f) if [Γn, Sn]→ [Γ, S] in Xd.

Proof. The condition [Γn, Sn] → [Γ, S] implies that Kn → K and hence afollows from equation (15) and b from Theorem 22, (14).

Corollary 24. Consider the free group Fd on d-generators g1, . . . , gd. For[Γ, S] in Xd define an epimorphism ϕ : Fd → Γ by setting ϕ(gi) = si ifS = (s1, . . . , sd).

a For every f ∈ L1(Fd), the following function is continuous:

T (f) : Xd → C defined by T (f)[Γ, S] = τNΓ(ϕ(f )).

b For every f ∈ L1(Fd)×, the function

D(f) : Xd → R>0 defined by D(f)[Γ, S] = detNΓ(ϕ(f))

is continuous.



Remarks The map T (f) depends only on the image of f in the quotient ofL1(Fd) by the subgroup generated by the commutators [g, h] = gh−hg. More-

over D(f) depends only on the image of f in the abelianization of L1(Fd)×.

Note that assertion b is not a formal consequence of a since there is no func-tional calculus in L1(Γ) allowing us to define the logarithm on all invertibleelements of the form f∗f .

4. Further problems

For a nonzero polynomial P in Z[X±11 , . . . , X±1

d ] it is well known that theMahler measure satisfies the inequality M(P ) ≥ 1. In fact m(P ) = logM(P )can be interpreted as the entropy of a suitable Zd-action and entropies arenon-negative, cp. [22]. For discrete groups Γ the question whether detNΓ f ≥ 1holds for f ∈ ZΓ has been much studied for the modified version of detNΓ wherethe zero eigenspace is discarded, cp. [24] for an overview. If r(f) is injectiveon L2(Γ), these results apply to detNΓ f itself. It is known for example thatfor such f and all residually amenable groups Γ we have detNΓ f ≥ 1. ForMahler measures the polynomials P with M(P ) = 1 are known by a theoremof Kronecker in the one-variable case and by a result of Schmidt in general,[26]. For them the above mentioned entropy is zero and this is significant forthe dynamics. Apart from Γ = Zd and finite groups Γ nothing seems to beknown about the following problem:

Question 25. Given a countable discrete group Γ, can one characterize theelements f ∈ ZΓ with detNΓ f = 1?

Even the case, where Γ is finitely generated and nilpotent would be inter-esting with the integral Heisenberg group as a starting point.

The polynomials P ∈ Z[X±11 , . . . , X±1

d ] with M(P ) = 1 are either units in

Z[X±11 , . . . , X±1

d ] or they have zeros on T d and hence are not invertible in L1.

For f ∈ Z[Zd] ∩ L1(Zd)× we therefore have detNZd f > 1 unless f is a unit inZ[Zd]. Is the same true in general?

Question 26. Given a countable discrete group Γ and an element f ∈ ZΓ ∩L1(Γ)× which does not have a left inverse in ZΓ, is detNΓ f > 1?

Remark If Γ is residually finite and amenable, the answer is affirmative. Thiswas shown in the proof of [10, Corollary 6.7] by interpreting log detNΓ f as anentropy and proving that the latter was positive. Note that if f does have aleft inverse in ZΓ i.e. gf = 1 for some g ∈ ZΓ we have (detNΓ g)(detNΓ f) = 1which implies that detNΓ f = 1 = detNΓ g if both determinants are ≥ 1.Incidentally, by a theorem of Kaplansky, NΓ and hence also the subrings CΓand L1(Γ) are directly finite, i.e. left units are right units and vice versa.

The last topic we want to mention concerns a continuity property. Answer-ing a question of Schinzel, Boyd proved the following result about the Mahlermeasure in [3]:

Theorem 27 (Boyd). For any Laurent polynomial P ∈ C[X±11 , . . . , X±1

d ] thefunction z 7→M(z − P ) is continuous in C.



The proof is based on an estimate due to Mahler which in turn uses Jensen’sformula.

Thus the question arises whether detNΓ(z − f) is a continuous function ofz ∈ C for f in CΓ. For A in NΓ the function ϕ(z) = log detNΓ(z − A) is asubharmonic function on C cp. [5] and in particular it is upper semicontinuous.For z /∈ σ(A) (or even for z outside the support of the Brown measure) thefunction ϕ(z) is easily seen to be continuous. If Γ is finite then detNΓ(z−f) =| det(z− r(f))|1/|Γ| is clearly continuous for z ∈ C. For the discrete Heisenberggroup Γ one may use formula (4) in [8] to get examples where detNΓ(z − f)can be expressed in terms of ordinary integrals. In all these cases one obtainsa continuous function of z if f is in CΓ.

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Received May 11, 2009; accepted May 20, 2009

Christopher DeningerWestfalische Wilhelms-Universitat Munster, Mathematisches InstitutEinsteinstr. 62, D-48149 Munster, GermanyE-mail: [email protected]




Equivariant K-theory of finite dimensional real

vector spaces

Siegfried Echterhoff and Oliver Pfante

(Communicated by Linus Kramer)


Abstract. We give a general formula for the equivariant complex K-theory K∗G

(V ) of afinite dimensional real linear space V equipped with a linear action of a compact group Gin terms of the representation theory of a certain double cover of G. Using this generalformula, we give explicit computations in various interesting special cases. In particular, asan application we obtain explicit formulas for the K-theory of C∗

r (GL(n, R)), the reducedgroup C*-algebra of GL(n, R).

1. Introduction

Let G be a compact group acting linearly on the real vector space V . Inthis paper we want to give explicit formulas for the complex equivariant K-theory K∗G(V ) depending on the action of the given group G on V . By use ofthe positive solution of the Connes-Kasparov conjecture in [5], this will alsoprovide explicit formulas for theK-theoryK∗(C

∗r (H)) of the reduced groupC∗-

algebra C∗r (H) for any second countable almost connected group H dependingon the action of the maximal compact subgroup G of H on the tangent spaceV = TeG(H/G).

If the action ofG on V is orientation preserving (which is always the case ifGis connected) and lifts to a homomorphism of G to Spin(V ) (or even Spinc(V )),we get the well-known answer from the equivariant Bott periodicity theorem.It implies that

K∗G(V ) ∼= K∗+dim(V )G (pt) =

⊕ρ∈ bG Z if ∗+ dim(V ) is even

0 if ∗+ dim(V ) is odd.

The obstruction for a linear action of G on V to lift to a homomorphism intoSpin(V ) is given by the Stiefel-Whitney class [ζ] ∈ H2(G,Z2), where we write

66 Siegfried Echterhoff and Oliver Pfante

Z2 := Z/2Z. This class ζ determines a central group extension

1 −−−−→ Z2 −−−−→ Gζ −−−−→ G −−−−→ 1.

If we denote by −1 the nontrivial element of Z2 ⊆ Gζ , then the irreducible

representations of Gζ can be divided into the disjoint subsets G+ζ and G−ζ with

G+ζ := ρ ∈ G : ρ(−1) = 1Vρ G−ζ := ρ ∈ G : ρ(−1) = −1Vρ.

It is then well known (e.g., see [5, §7]) that K∗G(V ) is a free abelian group

with one generator for every element ρ ∈ G−ζ . In particular, it follows that

the equivariant K-theory of V is always concentrated in dim(V ) mod 2 if theaction of G on V is orientation preserving.

The situation becomes more complicated if the action of G on V is notorientation preserving. As examples show, in this case nontrivial K-groupsmay appear in all dimensions. The situation has been studied in case of finitegroups by Karoubi in [8]. In this paper we use different methods to give ageneral description of K∗G(V ) which works for all compact groups G. We thenshow in several particular examples how one can extract explicit formulas fromour general result, also recovering the explicit formulas given by Karoubi in thecase of the symmetric group Sn acting on Rn by permuting the coordinates.

To explain our general formula we assume first that dim(V ) is even. Thenthere is a central extension

1 −−−−→ Z2 −−−−→ Pin(V )Ad−−−−→ O(V ) −−−−→ 1,

where Pin(V ) ⊂ ClR(V ) denotes the Pin group of V (see §2 for further detailson this extension). If ρ : G→ O(V ) is any continuous homomorphism, let

Gρ := (x, g) ∈ Pin(V )×G : Ad(x) = ρ(g).Then Gρ is a central extension of G by Z2 with quotient map q : Gρ → Ggiven by the projection to the second factor.

LetKρ ⊆ Gρ denote the pre-image of SO(V ) under the homomorphism ρq :Gρ → O(V ). Then Kρ = Gρ, if the action of G on V is orientation preserving,

and [Gρ : Kρ] = 2 otherwise. The group Gρ acts on Kρ by conjugation. Let

K−ρ = τ ∈ Kρ : τ(−1) = −1Vτ be the set of negative representations of Kρ. This set is invariant under the

action of Gρ. Write O1 as the set of all orbits of length one in K−ρ and O2 as

the set of all orbits of length two in K−ρ . We then get the following generalresult:

Theorem 1.1. Suppose that G is a compact group acting on the even dimen-sional real vector space V via the homomorphism ρ : G→ O(V ). Then, usingthe above notations, we have:

(i) If ρ(G) ⊆ SO(V ), then

KG0 (V ) ∼=

⊕

[τ ]∈bG−ρ

Z and KG1 (V ) = 0.


Equivariant K-theory of finite dimensional real vector spaces 67

(ii) If ρ(G) 6⊆ SO(V ), then

KG0 (V ) ∼=

⊕

[τ ]∈O2

Z and KG1 (V ) ∼=

⊕

[τ ]∈O1

Z.

The odd-dimensional case can easily be reduced to the even case by passingfrom V to V

⊕R together with Bott periodicity. The main tool for proving

the theorem is Kasparov’s KK-theoretic version of equivariant Bott periodicity,which provides a KKG-equivalence between C0(V ) and the complex Cliffordalgebra Cl(V ) (e.g. see [9, Theorem 7]). By the Green-Julg theorem, thisreduces everything to a study of KK(C,Cl(V ) ⋊ G), which then leads to theabove representation theoretic description of KG

∗ (V ).After having shown the above general theorem we shall consider various

special cases in which we present more explicit formulas. In particular we shallconsider the case of finite groups in §4 and the case of actions of O(n) in §5below. In particular, we show that for the canonical action of O(n) on Rn wealways get K1

O(n)(Rn) = 0 and

K0O(n)(R

n) ∼=

Z if n = 1⊕n∈N Z if n > 1

(see Example 4.2 for the case n = 1 and Theorem 5.7 for the case n > 1).Another interesting action of O(n) is the action by conjugation on the space Vnof symmetric matrices in M(n,R). By (the solution of) the Connes-Kasparovconjecture we have K∗

(C∗r (GL(n,R))

) ∼= K∗O(n)(Vn) which now allows us to

give explicit computations of these groups in all cases (see Theorem 5.9 below)In particular, in case n = 2 we get

K0

(C∗r (GL(2,R))

) ∼= Z and K1

(C∗r (GL(2,R))

) ∼=⊕

n∈N

Z,

which also shows that, for K∗G(V ), it is possible to have infinite rank in onedimension and finite rank in the other. While these results are stated andprepared in §5, some representation theoretical background and part of theproof are given in §6.

This paper is based on the Diplom thesis of the second named author writtenunder the direction of the first named author at the University of Munster. Theauthors are grateful to Linus Kramer for some useful discussions.

2. Some preliminaries on Clifford algebras

For the reader’s convenience, we recall in this section some basic facts onClifford algebras which will be used throughout this paper. For this let V bea fixed finite dimensional real vector space equipped with an inner product〈·, ·〉. We denote by ClR(V ) the real and by Cl(V ) = ClR(V )⊗R C the complexClifford algebras of V with respect to this inner product. Recall that ClR(V )is the universal algebra generated by the elements of V subject to the relation

v · v = −〈v, v〉1.



Every element of ClR(V ) is a finite linear combination of elements of the formv1 · v2 · · · vk with 0 ≤ k ≤ dim(V ) and there is a canonical Z2-grading onClR(V ) with grading operator

α : ClR(V )→ ClR(V ), α(v1 · · · vk) = (−1)kv1 · · · vk.We shall write ClR(V )0 and ClR(V )1 for the even and odd graded elementsof ClR(V ), respectively. We also have an involution on ClR(V ) given by(v1 · · · vk)∗ = (−1)kvk · · · v1. With this notation, the Pin group is definedas

Pin(V ) = x ∈ ClR(V ) : x∗x = 1 and xvx∗ ∈ V for all v ∈ V and Spin(V ) = Pin(V ) ∩ ClR(V )0, where we regard V as a linear subspace ofClR(V ) in the canonical way. Similar statements hold for the complex Clif-ford algebra Cl(V ) if we replace V by its complexification VC = V ⊗R C. Inparticular, we obtain the complex Pin group

Pinc(V ) = x ∈ Cl(V ) : x∗x = 1 and xvx∗ ∈ V for all v ∈ V ,and Spinc(V ) = Pinc(V ) ∩ Cl(V )0, where we regard V as a linear subspace ofCl(V ) via the inclusion x 7→ x⊗R 1 of ClR(V ) into Cl(V ). Note that this mapalso induces an inclusion ι : Pin(V )→ Pinc(V ).

If V is even dimensional with dimension 2n, then Cl(V ) is isomorphic to thefull matrix algebra M2n(C) and the grading on Cl(V ) is given by conjugationwith the element

J := e1e2 · · · e2n ∈ Spin(V ),

where e1, . . . , e2n is any given orthonormal base of V . Moreover, there is ashort exact sequence

(2.1) 1 −−−−→ Z2 −−−−→ Pin(V )Ad−−−−→ O(V ) −−−−→ 1,

where for x ∈ Pin(V ) the transformation Ad(x) ∈ O(V ) is defined byAd(x)(v) = xvx∗. The group Spin(V ) is the inverse image of SO(V ) underthe adjoint homomorphism. For more details on Clifford algebras and the Pingroups we refer to [2].

Notice that an analogue of the above extension is given in the odd-dimensional case by

(2.2) 1 −−−−→ Z2 −−−−→ Pin(V )A−−−−→ O(V ) −−−−→ 1,

where A : Pin(V ) → O(V ) denotes the twisted adjoint given by A(x)(v) =α(x)vx∗ . But this will not play a serious role in this paper.

Suppose now that G is a compact group and that ρ : G→ O(V ) is a linearrepresentation of G on V . Then ρ induces an action γ : G → Aut(Cl(V )) bydefining

γg(v1 · · · vk) = ρ(g)(v1) · · · ρ(g)(vk).Note that if x ∈ Pinc(V ) such that Ad(x) = ρ(g) for some g ∈ G, then

(2.3) γg(y) = xyx∗



for all y ∈ Cl(V ), which can be checked on the basic elements x = v1 · · · vk.Note that this action is compatible with the grading α = AdJ on Cl(V ).

If dim(V ) is even, we define

Gρ := (x, g) ∈ Pin(V )×G : Ad(x) = ρ(g).Then the kernel of the projection q : Gρ → G equals Z2 and we obtain a centralextension

(2.4) 1 −−−−→ Z2 −−−−→ Gρq−−−−→ G −−−−→ 1.

Let u : Gρ → Pin(V ), u(x, g) = x denote the canonical homomorphism. Thenit follows from (2.3) that

(2.5) γg(y) = u(x, g)yu(x, g)∗ for all y ∈ Cl(V ).

Note also that for each v ∈ Pinc(V ) we have the equation

JvJ∗ = α(v) = det(Ad(v))v

which follows from the fact that for v ∈ Pinc(V ) the transformation Ad v onV has determinant 1 if and only if x ∈ Cl(V )0. In particular we get

(2.6) Ju(x, g)J∗ = det ρ(g)u(x, g) for all (x, g) ∈ Gρ.If dim(V ) is odd, we consider the homomorphism ρ : G → O(V

⊕R) given

by ρ(g) =(ρ(g) 00 1

). We then put Gρ := Gρ. In this way we obtain a similar

central extension as in (2.4) in the odd-dimensional case.

3. The main result

Throughout this section we assume that ρ : G → O(V ) is a linear actionof the compact group G on the finite-dimensional real vector space V andwe let γ : G → Aut(Cl(V )) denote the corresponding action on the complexClifford algebra Cl(V ). Let us recall Kasparov’s KK-theoretic version of theBott periodicity theorem:

Theorem 3.1 ([9, Theorem 7]). Let ρ : G → O(V ) be as above. Then there

are classes α ∈ KKG0 (C0(V ),Cl(V )) and β ∈ KKG

0 (Cl(V ), C0(V )) which areinverse to each other with respect to the Kasparov product and therefore in-duce a KKG-equivalence between the graded C∗-algebra Cl(V ) and the triviallygraded algebra C0(V ).

From this theorem and the Green-Julg theorem (e.g., see [1, 20.2.7]), itfollows that

K∗G(V ) = KKG∗ (C, C0(V )) ∼= KKG

∗ (C,Cl(V )) ∼= KK∗(C,Cl(V ) ⋊γ G).

So in order to describe the K-theory groups K∗G(V ) it suffices to computethe groups KK∗(C,Cl(V ) ⋊γ G). Note that we use the KK-notation here andnot the notation K∗(Cl(V ) ⋊γ G), since it is important to keep in mind thatCl(V ) ⋊γ G is a graded algebra. Indeed, for any function f ∈ C(G,Cl(V )),



regarded as a dense subalgebra of Cl(V )⋊γG, the grading operator ǫ : Cl(V )⋊γ

G→ Cl(V ) ⋊γ G is given by

ǫ(f)(g) = α(f(g)),

where α : Cl(V )→ Cl(V ) denotes the grading of Cl(V ).In case where dim(V ) = 2n is even, we shall explicitly describe the crossed

product Cl(V ) ⋊γ G as a direct sum of full matrix algebras indexed by certainrepresentations of the compact group Gρ as defined in (2.4).

In general, we have Z2 = ±1 as a central subgroup of Gρ which givesus a distinct element −1 ∈ Gρ. We then write −g for (−1)g for all g ∈ Gρ.A function f ∈ C(Gρ) is said to be even (resp. odd), if f(−g) = f(g) (resp.f(−g) = −f(g)) for all g ∈ Gρ. The even functions can be identified withC(G) in a canonical way, and a short computation shows that the convolutionon C(Gρ) restricts to ordinary convolution on C(G) ⊆ C(Gρ). Similarly, theset of odd functions C(Gρ)

− is also closed under convolution and involution,and we shall write C∗(Gρ)

− for its closure in C∗(Gρ). With this notation weget

Lemma 3.2. The decomposition of C(Gρ) into even and odd functions inducesa direct sum decomposition

C∗(Gρ) = C∗(G)⊕

C∗(Gρ)−

with projections φ+ : C∗(Gρ) → C∗(G), φ− : C∗(Gρ) → C∗(Gρ)− given on

f ∈ C(Gρ) by

φ+(f)(g) =1

2

(f(g) + f(−g)

)and φ−(f)(g) =

1

2

(f(g)− f(−g)

).

There is a corresponding decomposition of Gρ as a disjoint union G∪ G−ρ with

G−ρ := τ ∈ Gρ : τ(−1) = −1Vτ.Proof. The proof is fairly straight-forward: If τ : Gρ → U(Vτ ) is any irreducibleunitary representation of Gρ, then τ(−1) commutes with τ(g) for all g ∈ Gρand since τ(−1)2 = 1Vτ , it follows from Schur’s lemma that either τ(−1) = 1Vτor τ(−1) = −1Vτ . In the first case, the representation factors through anirreducible representation of G via the quotient map q : Gρ → G. Thus we

obtain a decomposition of the dual Gρ into the disjoint union Gρ = G ∪ G−ρ .One easily checks that the even (resp. odd) functions on Gρ are annihilated by

(the integrated forms of) all elements of G−ρ (resp. G), which then implies that

this decomposition of Gρ corresponds to the above described decomposition ofC∗(Gρ).

Definition 3.3. A representation τ of Gρ is called negative if τ(−1) = −1Vτ .

The negative representations are precisely those unitary representation ofGρ which factor through C∗(Gρ)

−. In what follows next, we want to showthat in case where dim(V ) = 2n is even, there is a canonical isomorphism



Cl(V ) ⊗ C∗(Gρ)− ∼= Cl(V ) ⋊γ G. To prepare for this result we first note

that the tensor product Cl(V ) ⊗ C∗(Gρ)− can be realized as the closure of

the odd functions f ∈ C(Gρ,Cl(V )) in the crossed product Cl(V ) ⋊idGρ withrespect to the trivial action of Gρ on Cl(V ). The representations are givenby the integrated forms ϕ × σ of pairs of representations (ϕ, σ) on Hilbertspaces H such that ϕ is a ∗-representation of Cl(V ), σ is a negative unitaryrepresentation of Gρ, and

ϕ(x)σg = σgϕ(x) for all g ∈ Gρ, x ∈ Cl(V ).

For the following proposition recall that q : Gρ → G denotes the quotient map.

Proposition 3.4. Suppose that dim(V ) = 2n is even. Then there is a canon-ical isomorphism

Θ : Cl(V )⊗ C∗(Gρ)−∼=−→ Cl(V ) ⋊γ G

which sends an odd function f ∈ C(Gρ,Cl(V )) to the function Θ(f) ∈C(G,Cl(V )) ⊆ Cl(V ) ⋊γ G, given by

Θ(f)(q(g)) =1

2

(f(g)ug + f(−g)u−g

).

Under this isomorphism, a representation ϕ × σ of Cl(V ) ⊗ C∗(Gρ)− on a

Hilbert space H corresponds to the representation ϕ × τ of Cl(V ) ⋊γ G on Hwith τ : G→ U(H) given by

τ(q(g)) = ϕ(u∗g)σ(g).

Proof. For the proof we first inflate the action γ : G→ Aut(Cl(V )) to an actionγ : Gρ → Aut(Cl(V )) in the obvious way. It follows then from (2.5) that thisaction is implemented by the canonical homomorphism u : Gρ → Pin(V ) insuch a way that

γg = Adug

for all g ∈ Gρ. It follows that the crossed product Cl(V ) ⋊γ Gρ is isomorphicto Cl(V ) ⋊id Gρ ∼= C∗(Gρ) ⊗ Cl(V ) with isomorphism Φ : Cl(V ) ⋊id Gρ →Cl(V ) ⋊γ Gρ given on the dense subalgebra C(Gρ,Cl(V )) by

(Φ(f)

)(g) = f(g)ug.

On the other hand, we have a canonical surjective ∗-homomorphism

Ψ : Cl(V ) ⋊γ Gρ → Cl(V ) ⋊γ G

given on C(Gρ,Cl(V )) by Ψ(f)(q(g)) = 12 (f(g) + f(−g)).

We claim that the ∗-homomorphism Θ : Cl(V ) ⊗ C∗(Gρ) → Cl(V ) ⋊γ Ggiven by the composition

Cl(V )⊗ C∗(Gρ) ∼= Cl(V ) ⋊id GρΦ−−−−→ Cl(V ) ⋊γ Gρ

Ψ−−−−→ Cl(V ) ⋊γ G

factors through the desired isomorphism Cl(V ) ⊗ C∗(Gρ)− ∼= Cl(V ) ⋊γ G. Itis then clear that it is given by the formula as in the proposition.



For the proof of the claim it suffices to show that the map

Θ : (Cl(V ) ⋊γ G) → (Cl(V )⊗ C∗(Gρ)) , ϕ× τ 7→ (ϕ× τ) Θ

is injective with image (Cl(V )⊗ C∗(Gρ)−) .To see this let (ϕ, τ) be any covariant representation of

(Cl(V ), G, γ

)on the

Hilbert space H . Composing it with the quotient map Ψ gives the covariantrepresentation (ϕ, τ q) of

(Cl(V ), Gρ, γ

). This representation corresponds to

the representation (ϕ, σ) of(Cl(V ), Gρ, id

)with σ : Gρ → U(Vτ ) given by

σg = ϕ(ug) · τq(g).

To see this, we simply compute for given f ∈ C(Gρ,Cl(V )) the integrated form

ϕ× τ q(Φ(f)) =

∫

Gρ

ϕ(Φ(f)(g))τq(g) dg

=

∫

Gρ

ϕ(f(g)ug)τq(g) dg = ϕ× σ(f).

Since u−1 = −1 in Cl(V ), we see that σ(−1) = ϕ(−1)τ(1) = −1H , whichimplies that σ factors through a representation of C∗(Gρ)

−, and thereforeϕ × σ = (ϕ × τ) Θ is an irreducible representation of Cl(V ) ⊗ C∗(Gρ)

−.Conversely, assume that a representation ϕ×σ of Cl(V )⊗C∗(Gρ)− on a Hilbertspace H is given. Then one checks that (ϕ, τ) with τ(q(g)) = ϕ(u∗g)σ(g) is a

covariant representation of(Cl(V ), G, γ

)such that ϕ × σ = (ϕ × τ) Θ, and

the result follows.

Recall that by the Peter-Weyl Theorem the C*-algebra C∗(Gρ) of the com-pact group Gρ has a decomposition

C∗(Gρ) =⊕

τ∈ bGρ

End(Vτ )

with projection from C∗(Gρ) onto the summand End(Vτ ) given by f 7→ τ(f) =∫Gρf(g)τg dg. The above decomposition together with Proposition 3.4 induces

a decomposition

Cl(V ) ⋊γ G ∼=⊕

τ∈bG−ρ

Cl(V )⊗End(Vτ ).

We need to analyze the grading on the direct sum decomposition induced bythe grading ǫ of Cl(V ) ⋊γ G. Recall that the latter is given on functionsf ∈ C(G,Cl(V )) by

ǫ(f)(g) = α(f(g)) = Jf(g)J∗

with J = e1 · · · e2n and e1, . . . , e2n an orthonormal basis of V . If Θ isthe isomorphism of Proposition 3.4 we compute for any elementary tensor



x⊗ f ∈ Cl(V )⊗ C(Gρ)− ⊂ C(Gρ,Cl(V )) that

ǫ(Θ(x⊗ f))(q(g)) = JΘ(x⊗ f)(q(g))J∗

= J1

2

(xf(g)ug + xf(−g)u−g

)J∗

=1

2

(JxJ∗f(g)JugJ

∗ + JxJ∗f(−g)Ju−gJ∗)

= JxJ∗ det(ρ(q(g)))1

2

(f(g)ug + f(−g)u−g

)

= Θ(α(x) ⊗ (det ρ) · f)(q(g)),

where the second to last equation follows from (2.6). This shows that thegrading on Cl(V )⊗C∗(Gρ)− corresponding to ǫ is the diagonal grading givenby the standard grading α on Cl(V ) and the grading on C∗(Gρ)

− given onC(Gρ)

− via point-wise multiplication with the Z2-valued character

µ : Gρ → Z2, µ(g) = det ρ q(g).Now, for any function f ∈ C(Gρ) and τ ∈ Gρ we get

τ(µ · f)(g) =

∫

Gρ

µ(g)f(g)τg dg = (µ · τ)(f),

which implies that the corresponding grading on C∗(Gρ)− =

⊕τ∈ bG−

ρEnd(Vτ )

induces an inner automorphism on the block End(Vτ ) if τ ∼= µ·τ and intertwinesEnd(Vτ ) with End(Vµτ ) if µτ 6∼= τ .

Write Mτ := Cl(V ) ⊗ End(Vτ ). Then Mτ is isomorphic to a full matrixalgebra. If τ ∼= µ · τ , this summand of Cl(V ) ⊗ C∗(Gρ)

− is fixed by thegrading and Mτ is Morita equivalent (as graded algebra) to the trivially gradedalgebra Mτ . If µτ 6∼= τ , the grading intertwines Mτ with Mµτ and the directsum Mτ

⊕Mµτ is isomorphic to the algebra Mτ

⊕Mτ with the standard odd

grading given by (S, T ) 7→ (T, S). Thus Mτ

⊕Mµτ is isomorphic to Mτ ⊗Cl1,

where Cl1 = C⊕

C denotes the first Clifford algebra. We therefore obtain adecomposition

(3.1) Cl(V ) ⋊γ G ∼=(⊕

τ∈O1

Mτ

)⊕ ⊕

τ,µτ∈O2

Mτ ⊗Cl1

,

where O1 denotes the set of fixed points in G−ρ under the order-two transfor-mation τ 7→ µτ , and O2 denotes the set of orbits of length two under thisaction. Using this, it is now easy to prove:

Theorem 3.5. Let ρ : G → O(V ) be a linear action of the compact groupG on the finite-dimensional real vector space V and let γ : G → Aut(Cl(V ))

denote the corresponding action on Cl(V ). Let G−ρ , O1 and O2 be as above.Then

K∗G(V ) ∼= KK∗(C,Cl(V ) ⋊γ G) =

⊕τ∈O1

Z if ∗+ dim(V ) = 0 mod2⊕τ,µτ∈O2

Z if ∗+ dim(V ) = 1 mod2.



Proof. We first assume that dim(V ) = 2n is even. The isomorphism K∗G(V ) ∼=KK∗(C,Cl(V ) ⋊γ G) is Kasparov’s Bott periodicity theorem. The decomposi-tion of Cl(V ) ⋊γ G of (3.1) implies a decomposition

KK∗(C,Cl(V ) ⋊γ G) ∼= KK∗

(C,⊕

τ∈O1

Mτ

)⊕KK∗

(C,

⊕

τ,µτ∈O2

Mτ ⊗Cl1

)

= K∗

( ⊕

τ∈O1

Mτ

)⊕K∗+1

( ⊕

τ,µτ∈O2

Mτ

)

=

( ⊕

τ∈O1

K∗(Mτ )

)⊕( ⊕

τ,µτ∈O2

K∗+1(Mτ )

),

and the result follows from the fact that K0(M) = Z and K1(M) = 0 forany full matrix algebra M .

If dim(V ) is odd, we defined Gρ = Gρ where ρ : G → O(V⊕

R), ρ(g) =(ρ(g) 00 1

). Note that we haveK∗G(V ) ∼= K∗+1

G (V⊕

R) by Bott periodicity. Thusthe result follows from the even-dimensional case applied to the action ρ onV⊕

R.

Note that the homomorphism µ is trivial if and only if ρ takes image inSO(V ), i.e., the action of G on V is orientation preserving. In this case we get

O1 = G−ρ and O2 = ∅. Moreover, we then have

Gρ = (x, g) ∈ Spin(V )×G : Adx = ρ(g).This construction of Gρ makes also sense if the dimension of V is odd, and itthen coincides with the extension Gρ we obtain by passing to the action ρ ofG on V

⊕R. We leave the verification of this simple fact to the reader. We

therefore recover the following well-known result (e.g., see [5, §7]):

Corollary 3.6. Let ρ : G→ SO(V ) be an orientation preserving linear actionof the compact group G on the finite dimensional real vector space V . Then

K∗G(V ) ∼=⊕

τ∈ bG−ρ

Z if ∗+ dim(V ) = 0 mod 2

0 if ∗+ dim(V ) = 1 mod 2.

At this point it might be interesting to notice that for infinite compact

groups G the cardinality of G−ρ is always countably infinite if G is secondcountable. This follows from the next lemma.

Lemma 3.7. Let 1→ Z2 → H → G→ 1 be a central extension of an infinite

compact second countable group G by Z2. Then the set H− of equivalenceclasses of negative irreducible representations of H is countably infinite.

Proof. We decompose L2(H) as a direct sum L2(G)⊕L2(H)−, where we

identify L2(G) with the set of even functions in L2(H), and where L2(H)−

denotes the set odd functions in L2(H). Since G is not finite, both spacesare separable infinite dimensional Hilbert spaces. The regular representationλH : H → U(L2(H)) then decomposes into the direct sum λG

⊕λ−H , with



λ−H a negative representation of H . By the Peter-Weyl Theorem we get adecomposition

λG⊕

λ−H =⊕

τ∈ bH

dτ · τ,

where dτ denotes the dimension of τ and dτ · τ stands for the dτ -fold directsum of τ with itself. Since a direct summand of a negative (resp. positive)representation must be negative (resp. positive), we get the decomposition

λ−H =⊕

τ∈ bH−

dτ τ.

The result now follows from the fact that all representations τ in this decom-position are finite dimensional.

We proceed with a discussion of the special case, where the homomorphismρ : G→ O(V ) (resp. ρ : G→ O(V

⊕R) in case where dim(V ) is odd) factors

through a homomorphism v : G→ Pinc(V ) (resp. Pinc(V⊕

R)). We then saythat the action satisfies a Pinc-condition. If ρ(G) ⊆ SO(V ), this implies thatthe action factors through a homomorphism to Spinc(V ), in which case we saythat the action satisfies a Spinc-condition. Assume first that dim(V ) is even.It follows then from (2.3) that the corresponding action on Cl(V ) is given byγ = Ad v, which then implies that

Cl(V ) ⋊γ G ∼= Cl(V ) ⋊id G ∼= Cl(V )⊗ C∗(G)

with isomorphism Θ : Cl(V ) ⋊id G → Cl(V ) ⋊γ G given on functions f ∈C(G,Cl(V )) by (

Θ(f))(g) = f(g)vg.

Let ǫ be the grading on Cl(V ) ⋊γ G. On elementary tensors x⊗ f ∈ Cl(V )⊗C(G) ⊂ Cl(V )⊗ C∗(G) we compute

ǫ(Θ(x⊗ f)

)(g) = J(xf(g)vg)J

∗ = JxJ∗ det(ρ(g))f(g)vg

= Θ(α(x) ⊗ (det ρ) · f

)(g),

where the second to last equation follows from (2.6). So we see that if µ :G → Z2 denotes the character µ = det ρ then the grading on Cl(V ) ⋊γ Gcorresponds to the diagonal grading on Cl(V ) ⊗ C∗(G) given on the secondfactor by multiplication with the character µ. Passing to V

⊕R in case where

dim(V ) is odd, we now obtain the following theorem, where the proof proceedsprecisely as in Theorem 3.5:

Theorem 3.8. Assume that the linear action ρ : G→ O(V ) satisfies a Pinc-

condition as defined above. Let O1 and O2 denote the sets of orbits in G underthe order two transformation τ 7→ µτ with µ = det ρ : G→ Z2. Then

K∗G(V ) ∼= KK∗(C,Cl(V ) ⋊γ G) =

⊕τ∈O1

Z if ∗+ dim(V ) = 0 mod2⊕τ,µτ∈O2

Z if ∗+ dim(V ) = 1 mod2.



In particular, if ρ satisfies a Spinc-condition, then O1 = G and O2 = ∅ andthen

K∗G(V ) ∼= KK∗(C,Cl(V ) ⋊γ G) =

⊕τ∈ bG Z if ∗+ dim(V ) = 0 mod2

0 if ∗+ dim(V ) = 1 mod2.

Note that the above statements are different from the statement given inthe introduction, where we describe the K-groups in terms of the kernel Kρ of

the character µ : Gρ → Z2 and the action of Gρ on Kρ.In order to see that the above results can be formulated as in the introduc-

tion let us assume that G is any compact group, µ : G → Z2 is a nontrivialcontinuous group homomorphism, and K := kerµ ⊆ G. Then K is a normal

subgroup of index two in G and G acts on K by conjugation. This action is

trivial on K and therefore factors to an action of G/K ∼= Z2 on K. Thus the

G-orbits in K are either of length one or of length two.

Proposition 3.9. Let µ : G → Z2 and K = kerµ as above. Consider the

action of Z2 on G given on the generator by multiplying with µ and let Z2∼=

G/K act on K via conjugation. Then there is a canonical bijection

res : G/Z2 → K/Z2

which maps orbits of length one in G to orbits of length two in K and viceversa.

This proposition is well known to the experts and follows from basic rep-resentation theory. An elementary proof in case of finite groups G is given in[7, Theorem 4.2 and Corollary 4.3] and the same arguments work for generalcompact groups. Let g ∈ G \K be any fixed element. The basic steps for theproof are as follows:

• If τ is an irreducible representation of G, the restriction τ |K is eitherirreducible, or it decomposes into the direct sum σ

⊕g · σ for some

σ ∈ K with g · σ(k) = σ(g−1kg) for k ∈ K.• τ |K is irreducible if and only if τ 6∼= µτ .

• If τ |K ∼= σ⊕g · σ for some σ ∈ K, then σ 6∼= g · σ.

The map res : G/Z2 → K/Z2 of the proposition is then given by sending the

an orbit τ, µτ of length two in G to the orbit τ |K of length one in K

and an orbit τ of length one in G to the orbit σ, g · σ of length two in Kdetermined by τ |K ∼= σ

⊕g · σ. It follows from Frobenius reciprocity that this

map is onto.We now come back to the description of K∗G(V ):

Corollary 3.10. Suppose that G is a compact group and ρ : G → O(V ) isa linear action of G on the finite dimensional real vector space V . Assumethat µ = det ρ q : Gρ → Z2 is not trivial, i.e the action of G on V is notorientation preserving. Let Kρ = kerµ ⊆ G and let

K−ρ = σ ∈ Kρ : σ(−1) = −1Vσ.



Then K−ρ is invariant under the conjugation action of Gρ on Kρ. Let O1 and

O2 denote the set of orbits of length one or two in K−ρ . Then

K∗G(V ) ∼=⊕

σ∈O1Z if ∗+ dim(V ) = 1 mod2⊕

σ,gσ∈O2Z if ∗+ dim(V ) = 0 mod2.

Proof. We first note that the central subgroup Z2 of Gρ lies in the kernel of

µ, since µ factors through G. So the definition of K−ρ makes sense. Moreover,

since Z2 is central in Gρ, we get g(−1)g−1 = −1 for all g ∈ Gρ, which implies

that K−ρ is invariant under the conjugation action. The description of the

bijection res : Gρ/Z2 → Kρ/Z2 of Proposition 3.9 given above now implies

that it restricts to a bijection res− : G−ρ /Z2 → K−ρ /Z2, which maps orbitsof length one to orbits of length two and vice versa. The result then followsdirectly from Theorem 3.5.

In case where ρ : G → O(V ) satisfies a Pinc-condition as considered inTheorem 3.8, the same proof as for the above corollary together with Theorem3.8 gives

Corollary 3.11. Suppose that the linear action ρ : G → O(V ) satisfies aPinc-condition and assume that µ = det ρ : G → Z2 is nontrivial. Let K =

kerµ ⊆ G and let O1 and O2 denote the set of G-orbits in K of length oneand two, respectively. Then

K∗G(V ) ∼=⊕

σ∈O1Z if ∗+ dim(V ) = 1 mod2⊕

σ,gσ∈O2Z if ∗+ dim(V ) = 0 mod2.

4. Actions of finite groups

In this section we want to study the case of finite groups in more detail.This case was already considered by Karoubi in [8], but the methods used hereare different from those used by Karoubi. We first notice that it follows fromTheorem 3.5 that for actions of finite groups G, the K-theory groups K∗G(V )are always finitely generated free abelian groups. In what follows next we wantto give formulas for the ranks of these groups in terms of conjugacy classes.

Recall that for every finite group G the number |G| of (equivalence classes of)irreducible representations of G equals the number CG of conjugacy classes inG.

We first look at the case where the action ρ : G → O(V ) satisfies a Pinc-condition. Let µ = det ρ : V → Z2. If µ is trivial it follows from Theorem 3.8

that rank(K∗G(V )) = |G| = CG if ∗+ dim(V ) = 0 mod2 and rank(K∗G(V )) = 0otherwise.

If µ is nontrivial, let K = kerµ and let O1 and O2 denote the number of

G-orbits in K of length one and two, respectively. The numbers O1, O2 satisfythe equations

(4.1) O1 + 2O2 = |K| = CK and 2O1 +O2 = |G| = CG.



Indeed, the first equation follows from the obvious fact that O1 + 2O2 coin-cides with the number of irreducible representations of K and it follows fromProposition 3.9 that 2O1 +O2 coincides with the number of irreducible repre-sentations of G. By basic linear algebra the equations (4.1) have the uniquesolutions

(4.2) O1 =1

3

(2CG − CK) and O2 =

1

3(2CK − CG).

Combining all this with Corollary 3.11 implies

Proposition 4.1. Suppose that the linear action ρ : G → O(V ) of the finitegroup G satisfies a Pinc-condition. Then

(4.3) rank(K∗G(V )) =

CG if ∗+ dim(V ) = 0 mod2

0 if ∗+ dim(V ) = 1 mod2

if the action is orientation preserving, and

(4.4) rank(K∗G(V )) =

13

(2CK − CG) if ∗+ dim(V ) = 0 mod2

13 (2CG − CK) if ∗+ dim(V ) = 1 mod2

if the action is not orientation preserving, where K = g ∈ G : det(ρ(g)) = 1.

Example 4.2. Let G = Zm, the cyclic group of orderm, and let ρ : Zm → O(V )be a linear action of Zm on V . We claim that ρ automatically satisfies aPinc-condition. For this let g be a generator of Zm. By passing to V

⊕R

if necessary, we may assume without loss of generality that dim(V ) is even.Choose u ∈ Pinc(V ) such that Ad(c) = ρ(g). Then Ad(um) = ρ(gm) = ρ(e) =1V , and there exists λ ∈ T with um = λ1. Changing u into ζu, where ζ ∈ T isan mth root of λ, we obtain a well defined homomorphism v : Zm → Pinc(V )which sends g to ζu such that Ad v(gk) = ρ(gk) for all k ∈ Z. If the action ρtakes image in SO(V ) (which is automatic if m is odd), we get

K∗Zm(V ) ∼=

Zm if ∗+ dim(V ) = 0 mod 2

0 if ∗+ dim(V ) = 1 mod 2,

since CG = m. Assume now that m is even and the action is not orientationpreserving. Then the group K = ker(det ρ) is cyclic of order m

2 . With thevalues CG = m and CK = m

2 , Proposition 4.1 implies

K∗Zm(V ) ∼=

Zm/2 if ∗+ dim(V ) = 1 mod2

0 if ∗+ dim(V ) = 0 mod2.

As a particular example, consider the action of Z2 on R by reflection. Thenwe get

K0Z2

(R) = Z and K1Z2

(R) = 0.



We now study the general case. Consider the central extension 1 → Z2 →Gρ

q→ G→ 1 as in (2.4). As shown in the previous section we have a disjoint

decomposition Gρ = G−ρ ∪ G. We therefore get

|G−ρ | = |Gρ| − |G| = CGρ − CG.Thus, if the action ρ : G → SO(V ) is orientation preserving, it follows fromCorollary 3.6 that rank(K∗G(V )) = CGρ − CG if ∗ + dim(V ) = 0 mod 2 andrank(K∗G(V )) = 0 otherwise.

If ρ is not orientation preserving we obtain the nontrivial character µ : Gρ →Z2, µ = det ρ q. Let Kρ = kerµ and let O1 and O2 denote the number of

Gρ-orbits of length one and two in K−ρ , respectively. Similarly as for |G−ρ | wehave the formula

|K−ρ | = |Kρ| − |K| = CKρ − CK .Thus, as a consequence of Proposition 3.9 we see that

(4.5) O1 + 2O2 = |K−ρ | = CKρ − CK and 2O1 +O2 = |G−ρ | = CGρ − CG,As in the Pinc-case, these equations have unique solutions and, using Corollary3.10, we obtain

Proposition 4.3. Let ρ : G → O(V ) be a linear action of the finite group Gon V . Then

(4.6) rank(K∗G(V )) =

CGρ − CG if ∗+ dim(V ) = 0 mod2

0 if ∗+ dim(V ) = 1 mod2

if the action is orientation preserving, and(4.7)

rank(K∗G(V )) =

13

(2(CKρ − CK)− (CGρ − CG)

)if ∗+ dim(V ) = 0 mod2

13

(2(CGρ − CG)− (CKρ − CK)

)if ∗+ dim(V ) = 1 mod2

if the action is not orientation preserving.

Remark 4.4. Karoubi shows in [8] that for any linear action ρ : G→ O(V ) of afinite group G the ranks of K0

G(V ) and K1G(V ) can alternatively be computed

as follows: For any conjugacy class Cg in G let V g denote the fixed-point setof ρ(g) in V . This space is ρ(h)-invariant for any h in the centralizer Cg ofg, and therefore Cg acts linearly on V g for all g in G. With these facts inmind, Karoubi denotes a conjugacy class Cg oriented, if the action of Cg onV g is oriented and Cg is called even (resp. odd) if dim(V g) is even (resp. odd).He then shows in [8, Theorem 1.8] that the rank of K0

G(V ) (resp. K1G(V ))

equals the number of oriented even (resp. odd) conjugacy classes in G. Itseems not obvious to us that Karoubi’s result coincides with the result givenin Proposition 4.3 above.

In what follows we want to apply our results to give an alternative proofof the formulas for the ranks of K0

Sn(Rn) and K1

Sn(Rn), as given by Karoubi

in [8, Corollary 1.9], where the symmetric group Sn, for n ≥ 2, acts on Rn



by permuting the standard orthonormal base e1, . . . , en. Let ρ : Sn → O(n)denote the corresponding homomorphism. It is clear that the inverse imageof SO(n) is the alternating group An. To simplify notation, we shall write Snand An for the groups (Sn)ρ and (An)ρ. Thus it follows from Proposition 4.3that(4.8)

rank(K∗Sn(Rn)) =

13

(2(CAn − CAn)− (CSn − CSn)

)if ∗+ n = 0 mod2

13

(2(CSn − CSn)− (CAn − CAn)

)if ∗+ n = 1 mod2

Thus, to get explicit formulas we need to compute the numbers CSn−CSn andCAn − CAn . For this we use the following general observations: If

1 −−−−→ Z2 −−−−→ Gq−−−−→ G −−−−→ 1

is any central extension of G by Z2, the inverse image q−1(Cg) of a conjugacy

class Cg in G is either a conjugacy class in G itself, or it decomposes into twodisjoint conjugacy classes of the same length in G. If t ∈ G such that q(t) = g,

the second possibility happens if and only if t is not conjugate to −t in G (e.g.,see [7, Theorem 3.6]). Thus, if Cdec

G denotes the number of conjugacy classes

in G which decompose in G, the number CG of conjugacy classes in G is equal

to CG + CdecG , and hence

CG − CG = CdecG .

Now, for the groups G = Sn and K = An the numbers CdecSn

and CdecAn

havebeen computed explicitly in [7, Theorem 3.8 and Corollary 3.10]:

Proposition 4.5. For each n ≥ 2 let an (resp. bn) denote the number of allfinite tuples of natural numbers (λ1, . . . , λm) such that 1 ≤ λ1 < λ2 < · · · <λm,

∑mi=1 λi = n, and the number of even entries λi is even (resp. odd). Then

CdecSn = an + 2bn and Cdec

An = 2an + bn.

We should note that the definitions of Sn and An considered in [7] areslightly different from ours, but a study of the proofs of Theorem 3.8 andCorollary 3.10 in that paper shows that the arguments apply step by step toour situation. As a consequence we get

Corollary 4.6. Let ρ : Sn → O(n) be as above. Then

K∗Sn(Rn) =

Zan if ∗+ n = 0 mod2Zbn if ∗+ n = 1 mod2

.

Remark 4.7. In [8, Corollary 1.9], Karoubi gives the formulas

K0Sn(R

n) = Zpn and K1Sn(R

n) = Zin

where pn (resp. in) denotes the number of partitions n =∑mi=1 λi with 1 ≤

λ1 < · · · < λm and m = 2k even (resp. m = 2k + 1 odd). One checks that



the numbers pn and in are related to the numbers an and bn, as defined in thecorollary above, by the equations:

a2n+1 = i2n+1 b2n+1 = p2n+1

a2n = p2n b2n = i2n,

and hence Karoubi’s formula coincides with ours. We give the argument forthe equation a2n+1 = i2n+1; the other equations can be shown similarly. Solet n ∈ N and let (λ1, . . . , λm) be a partition of 2n + 1 as in the definitionof a2n+1, i.e., there is an even number 2r of even entries λi in this partition.Since λ1 + · · · + λm = 2n + 1 is odd, it follows that there is an odd numberof odd entries λj in the partition. Thus m = 2k + 1 is odd. Conversely, ifm = 2k+1 is odd, the fact that λ1 + · · ·+λm = 2n+1 is odd implies that thenumber l of odd entries λj must be odd, and then the number m − l of evenentries must be even.

If we restrict the action of Sn to the alternating group An, we obtain theformulas

K∗An(Rn) =

Z2an+bn if ∗+n = 0 mod2

0 if ∗+n = 1 mod2,

since this action is orientation preserving.

5. Actions of O(n)

In this section we want to study the K-theory groups K∗O(n)(V ) for linear

actions ρ : O(n) → O(V ) of the orthogonal group O(n) on an arbitrary realvector space V . We are in particular interested in the canonical action of O(n)on V = Rn and in the action of O(n) on the space Vn of all symmetric matricesin Mn(R), with action given by conjugation. The study of the latter will allowus to compute explicitly the K-theory groups of the reduced group C*-algebraC∗r (GL(n,R)) of the general linear group GL(n,R) via the positive solution ofthe Connes-Kasparov conjecture.

Recall that O(n) ∼= SO(n) × Z2 if n is odd, with −I ∈ O(n) the generatorof Z2 (in what follows we denote by I the unit matrix in O(n) ⊆ Mn(R)and we denote by 1 the unit in Spin(n) ⊆ ClR(n)). If n is even, we haveO(n) ∼= SO(n) ⋊ Z2, the semi-direct product of SO(n) with Z2, where thegenerator of Z2 can be chosen to be the matrix g := diag(−1, 1, . . . , 1) ∈ O(n)acting on SO(n) by conjugation.

Given a representation ρ : O(n) → O(V ), we need to describe the groupO(n)ρ and its representations. For this we start by describing all possiblecentral extensions of O(n) by Z2. Indeed, we shall see below that for any fixedn ≥ 2 there are precisely four such extensions

1 −−−−→ Z2 −−−−→ Gniq−−−−→ O(n) −−−−→ 1,

i = 0, . . . , 3. To describe them, we let Kni denote the inverse image of SO(n) in

Gni for i = 0, . . . , 3. This is a central extension of SO(n) by Z2 and therefore theKni are either isomorphic to the trivial extension SO(n)×Z2 or the nontrivial



extension Spin(n). Using this, the extensions Gni , 2 ≤ n ∈ N, i = 0, . . . , 3 aregiven as follows:If n = 2m+ 1 is odd, then

(O1) there are two extensionsGn0 andGn1 such thatKn0 = Kn

1 = SO(n)×Z2:the trivial extension Gn0 = O(n) × Z2 and the nontrivial extensionGn1 = SO(n) × Z4, with central subgroup Z2 being the order-twosubgroup of Z4.

(O2) There are two extensions Gn2 , Gn3 such that Kn

2 = Kn3 = Spin(n). To

characterize them let x ∈ Gni such that q(x) = −I ∈ O(n). Thenx2 = 1 for x ∈ Gn2 and x2 = −1 for x ∈ Gn3 . We then have Gn2

∼=Spin(n)× Z2 with x a generator for Z2 and Gn3

∼= (Spin(n)× Z4)/Z2

with respect to the diagonal embedding of Z2 into Spin(n)×Z4. Thecentral subgroup Z2 is given by (the image of) the order-two subgroup±1 ⊆ Spin(n).

If n = 2m is even, then

(E1) there are two extensionsGn0 andGn1 such thatKn0 = Kn

1 = SO(n)×Z2:the trivial extension Gn0 = O(n) × Z2 and the nontrivial extensionGn1 = SO(n) ⋊ Z4, with action of Z4 on SO(n) given on the generatorby conjugation with g = diag(−1, 1, . . . , 1), and the central subgroupZ2 of SO(n) ⋊ Z4 is given by the order-two subgroup of Z4.

(E2) There are two extensions Gn2 , Gn3 such that Kn

2 = Kn3 = Spin(n).

If x ∈ Gni such that q(x) = diag(−1, 1, . . . , 1), then x2 = 1 in casex ∈ Gn2 and x2 = −1 in case x ∈ Gn3 . Then Gn2

∼= Spin(n) ⋊ Z2 withZ2 generated by x and Gn3

∼= (Spin(n)⋊Z4)/Z2 where Z4 is generatedby an element x ∈ Spin(n)⋊Z4 which acts on Spin(n) by conjugationwith x, and Z2 is embedded diagonally into Spin(n)⋊Z4 as in the oddcase. The central copy of Z2 is given by (the image of) the order-twosubgroup ±1 ⊆ Spin(n).

Although we are convinced that this description of the central extensionsof O(n) by Z2 is well-known, we give a proof since we didn’t find a directreference:

Proposition 5.1. For any fixed n ≥ 2 the above described extensions are, upto isomorphism of extensions, the only central extensions of O(n) by Z2.

Proof. Recall first that the set of isomorphism classes of central extensions ofany given group H by Z2 forms a group E(H,Z2) with group operation givenas follows: if

1→ Z2 → Gq→ H → 1 and 1→ Z2 → G′

q′→ H → 1

are central extensions of H by Z2, then the product is given by the (isomor-phism class) of the extension

(5.1) 1 −→ Z2ι−→ G ∗G′ q′′−→ H −→ 1,



where G ∗ G′ = (x, x′) ∈ G × G′ : q(x) = q′(x′)/Z2 with respect to thediagonal embedding of Z2 into G×G′. The central copy of Z2 in G∗G′ can betaken as the image in G ∗G′ of the central copy of Z2 in either G or G′. It iswell known that E(SO(n),Z2) ∼= Z2 with nontrivial element given by Spin(n).

Suppose now that 1 → Z2 → Gq→ O(n) → 1 represents an element in

E(O(n),Z2). It restricts to a representative 1 → Z2 → K → SO(n) → 1in E(SO(n),Z2) with K := q−1(SO(n)). This restriction procedure inducesa homomorphism of E(O(n),Z2) to E(SO(n),Z2). Therefore, given any fixedextension G which restricts to K = Spin(n), then all other extension whichrestrict to Spin(n) are given as products G∗G′ where G′ is an extension whichrestricts to K ′ = SO(n) × Z2. In particular, if we can show that there areonly two extensions which restrict to SO(n)×Z2, then there are also only twoextensions which restrict to Spin(n). Since the ones given in the above list areobviously nonisomorphic (as extensions), the list must be complete.

So let G′ be any extension which restricts to K ′ = SO(n)×Z2. We show thatit equals Gn0 or Gn1 described above. Suppose first that n = 2m is even. Choosex ∈ G′ such that q(x) = g := diag(−1, 1, . . . , 1) ∈ O(n). Then q(x2) = I andhence x2 = ±1. We claim that

x(h, ǫ)x−1 = (ghg−1, ǫ)

for all (h, ǫ) ∈ SO(n) × Z2. To see this, note first that q(x(h, ǫ)x−1

)=

g(q(h, ǫ)

)g−1 = ghg−1, which implies that x(h, ǫ)x−1 = (ghg−1, ǫ′) for some

ǫ′ ∈ Z2. To see that ǫ′ = ǫ we simply observe that the map SO(n) → Z2

which sends h ∈ SO(n) to the projection of x(h, 1)x−1 to Z2 is a continuousgroup homomorphism, and hence trivial. This implies 1′ = 1 and then also(−1)′ = −1.

If x2 = 1, it follows that G = (SO(n) × Z2) ⋊ 〈x〉 = (SO(n) ⋊ 〈g〉) ×Z2 = O(n) × Z2 = Gn0 . If x2 = −1, we obtain a surjective homomorphismϕ : (SO(n) × Z2) ⋊ 〈x〉 → G given by ϕ

(xj , (g,±1)

)= xj(g,±1) with kernel

generated by the order-two element (x2,−1). This implies G = Gn1 .A similar but easier argument applies in the case where n = 2m+ 1 is odd.

We omit the details.

Remark 5.2. As we saw in Section 2 (e.g. see (2.1) and (2.2)) the group Pin(n)is a central extension of O(n) by Z2 which restricts to Spin(n). Thus the othersuch extension is given by the product Pin(n) ∗Gn1 . To see whether Pin(n) isthe group Gn2 or the group Gn3 , we need to identify an inverse image x ∈ Pin(n)of the matrix −I if n = 2m + 1 is odd, or of g = diag(−1, 1, . . . , 1) ∈ O(n) ifn = 2m is even.

Indeed, if e1, . . . , en denotes the standard orthonormal base of Rn, then itfollows from the basic relations in in ClR(n) that x = ±e1 · · · en if n = 2m+ 1is odd (with respect to extension (2.2)) and x = ±e2 · · · en if n = 2m is even.



In the first case we get

x2 = (−1)n(n+1)

2 = (−1)(m+1)(2m+1) =

−1 if m is even

1 if m is odd,

and in the second case we get

x2 = (−1)n(n−1)

2 = (−1)m(2m−1) =

1 if m is even

−1 if m is odd.

In what follows we need to understand the conjugation action of Gni on Kni .

Note that in all cases we can identify Gni /Kni with Z2.

Lemma 5.3. Let n ≥ 2, let 1→ Z2 → Gq→ O(n)→ 1 be any central extension

of O(n) by Z2 and let K = q−1(SO(n)). Then

(i) If n is odd, the conjugation action of G/K on K is trivial.

(ii) If n is even, and if K = SO(n) × Z2, the action of G/K on K =

SO(n)× Z2 is given by the conjugation action of O(n)/ SO(n) on thefirst factor and the trivial action on the second. If K = Spin(n), the

action of G/K on Spin(n) coincides with the conjugation action of

Pin(n)/ Spin(n) on Spin(n).

Proof. The first assertion follows directly from the description of the groupsGni in case where n is odd. So assume now that n is even and K = SO(n)×Z2.Let x ∈ G with q(x) = g := diag(−1, 1 . . . , 1). It is shown in the proof ofProposition 5.1 that the conjugation action of x on SO(n) × Z2 is given byconjugation with g ∈ O(n) in the first factor and the trivial action in thesecond factor. This proves the first assertion in (ii).

So assume now that K = Spin(n). Then G = Pin(n) or G = Pin(n) ∗ Gn1 .The result is clear in the first case. So let G = Pin(n) ∗Gn1 . If y = e2 · · · en ∈Pin(n) and x1 ∈ Gn1 with q1(x1) = g, then x = [y, x1] is an inverse image of gin G. The group Spin(n) then identifies with K ⊆ G via the embedding

ϕ : Spin(n)→ Pin(n) ∗Gn1 , z 7→ [z, (Ad(z), 1)].

Conjugating [z, (Ad(z), 1)] by [y, x1] provides [yzy−1, x1(Ad(z), 1)x−11 ] =

[yzy−1, (gAd(z)g−1, 1)] = ϕ(yzy−1), which finishes the proof.

Suppose now that ρ : O(n)→ O(V ) is any linear action of O(n) on a finitedimensional real vector space V . In all cases the group O(n)ρ must be one ofthe groups Gn0 , . . . , G

n3 . If we fix n ≥ 2, we have the following possibilities for

the computation of the K-theory groups K∗O(n)(V ):

The orientation preserving case: If the action of O(n) on V is ori-entation preserving, then Corollary 3.6 implies that

(5.2) K∗O(n)(V ) ∼=⊕

σ∈cGni− Z if ∗+ dim(V ) is even,

0 if ∗+ dim(V ) is odd.



Note that although we have four different possibilities Gn0 , . . . , Gn3 for the

groups O(n)ρ, it follows from Lemma 3.7 that the cardinality of Gni−

is alwayscountably infinite. Thus the isomorphism class of K∗O(n)(V ) only depends on

whether dim(V ) is even or odd.

The non-orientation preserving case: If the action of O(n) is notorientation preserving, the question which group out of the list Gn0 , . . . , G

n3

we get for O(n)ρ becomes more interesting (at least if n is even). In fact, ifi ∈ 0, . . . , 3 is such that O(n)ρ ∼= Gni as central extension of O(n) by Z2, itfollows from Corollary 3.10 that

(5.3) K∗O(n)(V ) ∼=⊕

σ∈O1Z if ∗+ dim(V ) is odd,⊕

σ,gσ∈O2Z if ∗+ dim(V ) is even,

where O1 and O2 denote the numbers of orbits of length one or two in the set

K− of negative representations of K := Kni under the conjugation action of

Gni . So in order to get the general picture, we need to study the cardinalitiesof the sets O1 and O2 in the four possible cases. We actually get differentanswers depending on whether n is even or odd:

The odd case n = 2m + 1: In this case Lemma 5.3 implies that the

action of Gni on K− is trivial in all cases. Thus, from the above formula weget

(5.4) K∗G(V ) ∼=⊕

σ∈ bK− Z if ∗+ dim(V ) is odd

0 if ∗+ dim(V ) is even.

As in the orientation preserving case, it follows from Lemma 3.7 that the

cardinality of K− is always countably infinite.

The even case n = 2m: Let G = O(n)ρ and K = q−1(SO(n)). If

K = SO(n) × Z2 we get K− = SO(n) × µ ∼= SO(n), where µ denotes thenontrivial character of Z2, and it follows from Lemma 5.3 that the action of

G/K on K− is given by the conjugation action of O(n)/ SO(n) on SO(n) ∼= K.Thus, the orbit sets O1 and O2 can be identified with the O(n)-orbits of length

one and two in SO(n) and the K-theory groups in the cases O(n)ρ = Gn0 andO(n)ρ = Gn1 are the same (up to isomorphism).

In case K = Spin(n) it follows from Lemma 5.3 that the action of G/K onSpin(n) coincides with the conjugation action of Pin(n)/ Spin(n) on Spin(n).

Thus, to compute the sets O1 and O2 in the K-theory formula (5.3), we mayassume without loss of generality that O(n)ρ = Pin(n).

In view of the above discussions, it is desirable to find an easy criterionfor the group K ⊆ O(n)ρ being isomorphic to SO(n) × Z2 or not. It is clearthat this is the case if and only if the restriction ρ : SO(n) → SO(V ) of



the given action ρ : O(n) → O(V ) is spinor in the sense that there exists ahomomorphism ρ : SO(n) → Spin(V ) such that ρ = q ρ with q : Spin(V ) →SO(V ) the quotient map. We believe that the following result can also bededuced from the results in [6] (we are grateful to Linus Kramer for pointingout this reference). We give an easy direct argument below:

Proposition 5.4. Let n ≥ 2 and let ρ : SO(n) → SO(V ) be any linearaction of SO(n) on the finite dimensional real vector space V . Let h :=diag(−1,−1, 1, . . . , 1) ∈ SO(n) and let V − = v ∈ V : ρ(h)v = −v denotethe eigenspace for the eigenvalue −1 of ρ(h). Then ρ is spinor if and only ifdim(V −) = 4k for some k ∈ N0.

Proof. We first need to know that, given a representation ρ : SO(n)→ SO(V ),there always exists a representation ρ : Spin(n) → Spin(V ) such that thediagram

(5.5)

Spin(n)ρ−−−−→ Spin(V )

q=Ad

yyq=Ad

SO(n) −−−−→ρ

SO(V )

commutes. In case n > 2 this follows from the universal properties of theuniversal covering Spin(n) of SO(n). In case n = 2, the groups Spin(2) andSO(2) are both isomorphic to the circle group T with covering map q : T→ T,z 7→ z2. The image ρ(SO(2)) lies in a maximal torus T ⊆ SO(V ) and there is a

maximal Torus T in Spin(V ) which projects onto T via a double covering map

q : T → T . Thus the problem reduces to the problem whether there exists amap ρ : T→ T which makes the diagram

Tρ−−−−→ T

q

yyq

T −−−−→ρ

T

commute. It is straightforward to check that this is always possible.It follows from (5.5) that there exists a lift ρ : SO(n)→ Spin(n) for ρ if and

only

±1 = ker(q : Spin(n)→ SO(n)

)⊆ ker ρ.

So we simply have to check whether ρ(−1) = 1 or not. For this let e1, . . . , endenote the standard orthonormal base of Rn. Then the product e1e2 ∈ Spin(n)projects onto h and (e1e2)

2 = −1. Since ρ(h)2 = ρ(h2) = 1 we see that Vdecomposes into the orthogonal direct sum V +

⊕V − with V + and V − the

eigenspaces for ±1 of ρ(h). Since det(ρ(h)) = 1, it follows that l = dim(V −)is even. If V − = 0 we have ρ(h) = 1, which implies ρ(e1e2) = ±1 and henceρ(−1) = ρ((e1e2)

2) = 1.



If V − 6= 0 let v1, . . . vl be any orthonormal base for V −. It then followsfrom the relations in Cl(V ) that the element

y := v1 · · · vl ∈ Spin(V )

projects onto ρ(h) ∈ SO(V ). This implies that ρ(e1e2) = ±y and hence that

ρ(−1) = ρ((e1e2)2) = y2 = (v1 · · · vl)2 = (−1)

l(l+1)2 .

Since l is even, we get l = 2m for some m ∈ N and then

ρ(−1) = (−1)m(2m+1) =

1 if m is even

−1 if m is odd.

This finishes the proof.

The only problem which now remains for the general computation ofK∗O(n)(V ) is the problem of computing explicitly the orbit sets O1 and O2

which appear in formula (5.3) in the case where n is even (as observed above,we always have O1 countably infinite and O2 = ∅ if 3 ≤ n = 2m + 1 isodd). In order to give the complete picture, we now state the general result,although we postpone the proof for the case n > 2 to §6 below:

Theorem 5.5. Suppose that ρ : O(n)→ O(V ) is a non-orientation preservingaction of O(n) on V with n = 2m even. Then the following are true:

(i) If the restriction ρ : SO(n)→ SO(V ) is spinor, then O1 consists of asingle point if n = 2 and O1 is countably infinite if n > 2. The setO2 is always countably infinite.

(ii) If ρ : SO(n)→ SO(V ) is not spinor, then O1 = ∅ and O2 is countablyinfinite.

Combining this result with (5.3) immediately gives

Corollary 5.6. Suppose that ρ : O(n)→ O(V ) is a non-orientation preservingaction of O(n) on V with n = 2m even. Then

K0O(n)

∼=⊕

n∈N

Z and K1O(n)(V ) ∼=

Z if n = 2,⊕

n∈N Z if 2 < n = 2m

if the restriction ρ : SO(n)→ SO(V ) is spinor. Otherwise we get

K0O(n)(V ) ∼=

⊕

n∈N

Z and K1O(n) = 0.

This corollary together with the discussions on the odd case implies

Theorem 5.7. Let O(n) act on Rn by matrix multiplication. Then

K0O(n)(R

n) =⊕

k∈N

Z and K1O(n)(R

n) = 0

for all n ∈ N with n ≥ 2.



Proof. Since the action is not orientation preserving and the restriction ofid : O(n) → O(n) to SO(n) is not spinor (which is an easy consequence ofProposition 5.4), the result follows from formula (5.4) in case where n is odd,and from Corollary 5.6 if n is even.

The case n = 2 of Theorem 5.5 is quite easy and has to be done separately,since the general methods used for n > 2 in §6 below will not apply to thiscase. So we do the case n = 2 now:

Proof of Theorem 5.5 in case n = 2. As usual, let K = q−1(SO(2)) denote theinverse image of SO(2) in O(2)ρ. If ρ : SO(2) → SO(V ) is spinor, we haveK = SO(2)× Z2. Otherwise we have K = Spin(2).

The case K = SO(2) × Z2: It follows from Lemma 5.3 that in this case the

sets O1 and O2 can be identified with the sets of O(2)-orbits in SO(2) of lengthone and two, respectively. Writing

SO(2) =gα =

(cos(α) sin(α)− sin(α) cos(α)

): α ∈ [0, 2π)

we have SO(2) = χk : k ∈ Z with χk(gα) = eikα. The action of O(2) on

SO(2) is given by conjugation with g = diag(−1, 1). Since ggαg−1 = g−α we

get g · χk = χ−k, which implies that

O1 = χ0 and O2 = χk, χ−k : k ∈ N.The case K = Spin(2): In this case the sets O1 and O2 can be identified

with the sets of Pin(2)-orbits in Spin(2)−

of length one and two, respectively.Recall that Spin(2) can be described as

Spin(2) = x(α) := cos(α)1 + sin(α)e1e2 : α ∈ [0, 2π) ⊆ ClR(2).

Then Spin(2) = χk : k ∈ Z with χk : Spin(2) → T, χk(x(α)) = eikα. It

follows that Spin(2)−

= χ2m+1 : m ∈ Z. The action of Pin(2) on Spin(2) isgiven by conjugation with x = e2. A short computation shows that

e2x(α)e∗2 = −e2x(α)e2 = x(−α)

which implies that x · χk = χ−k for all k ∈ Z. We therefore get x · χ2m+1 =

χ−2m−1 6= χ2m+1 for all χ2m+1 ∈ Spin(2)−

. Thus

O1 = ∅ and O2 = χ2m+1, χ−2m+1 : m ∈ N.

We close this section with another interesting application of our main results.Recall that for a locally compact group H , the reduced group C∗-algebraC∗r (H) is the closure of λ(L1(H)) ⊆ B(L2(H)), where

λ : L1(H)→ B(L2(H)), λ(f)ξ = f ∗ ξdenotes the left regular representation ofH . IfH is almost connected, it followsfrom the positive solution of the Connes-Kasparov conjecture, that there is a



(more or less) canonical isomorphism K∗(C∗r (H)) ∼= K∗G(V ), where G ⊆ Hdenotes the maximal compact subgroup of H and V = TeG(H/G) denotes thetangent space of H/G at the trivial coset eG = G. The action of G on Vis given by the differential of the left translation action of G on the manifoldH/G (see [5, §7]).

In case where H = GL(n,R), the maximal compact subgroup is O(n). IfVn = A ∈ M(n,R) : A = At denotes the space of symmetric matrices inMn(R), we have the well-known diffeomorphism

Vn ×O(n)→ GL(n,R), (A, g) 7→ exp(A)g,

with exp(A) =∑∞

n=01n!A

n the usual exponential map. Composing exp withthe quotient map GL(n,R)→ GL(n,R)/O(n) provides a diffeomorphism exp :Vn → GL(n,R)/O(n). We then get exp(gAg−1) = g · exp(A) and it followsfrom the above discussion that

(5.6) K∗(C∗r (GL(n,R))

) ∼= K∗O(n)(Vn)

for all n ≥ 2, where the action of O(n) on Vn is given by the representation

ρ : O(n)→ O(Vn), ρ(g)A = gAg−1

for all g ∈ O(n), A ∈ Vn ⊆M(n,R).

Lemma 5.8. Let ρ : O(n)→ O(Vn) be as above. Then

(i) ρ is orientation preserving if and only if n is odd.(ii) The restriction ρ : SO(n)→ SO(Vn) is spinor if and only if n is even.

Proof. Let Eij : 1 ≤ i ≤ j ≤ n denote the standard basis of Vn, i.e., Eij hasentry 1 at the ij-th and ji-th place, and 0 entries everywhere else. Conjugationwith g = diag(−1, 1, . . . , 1) ∈ O(n) maps E1j to −E1j for all j > 1 and fixesall other Eij ’s. It thus follows that det(ρ(g)) = (−1)n−1, which shows that ρis orientation preserving if and only if n is odd.

For the proof of (ii) we use Proposition 5.4: let h = diag(−1,−1, 1 . . . , 1) ∈SO(n). Then conjugation with h maps Eij to −Eij for all i = 1, 2 and j > 2and fixes all other Eij . Thus Eij : i = 1, 2, j > 2 forms a base for V −n , theeigenspace of ρ(h) for the eigenvalue −1. We therefore get l := dim(V −n ) =2(n− 2). This is a multiple of 4 if and only if n is even.

Theorem 5.9. If n = 2m+ 1 is odd, then

K∗(C∗r (GL(n,R))

) ∼=⊕

n∈N Z if ∗+m is odd

0 if ∗+m is even.

If n = 2m ≥ 4 is even, we get

K0(C∗r (GL(n,R)) ∼=

⊕

n∈N

Z ∼= K1

(C∗r (GL(n,R))

),

and for n = 2 we get

K0

(C∗r (GL(2,R))

) ∼= Z and K1

(C∗r (GL(2,R))

) ∼=⊕

n∈N

Z.



Proof. We use formula (5.6). If n = 2m + 1 is odd, the result then followsdirectly from formula (5.2) together with Lemma 5.8 above and the fact that

dim(Vn) = n(n+1)2 = (2m+ 1)(m+ 1) is even if and only if m is odd.

If n is even, it follows from Lemma 5.8 above that the action of O(n) on Vnis not orientation preserving and the restriction of ρ to SO(n) is spinor. Thusthe result follows from Corollary 5.6.

6. Orbits in Spin(n)−

and SO(n)

In this section we want to provide the theoretical background to completethe proof of Theorem 5.5. We need to compute the cardinalities for the orbit

sets O1 and O2 in Spin(m)−

under the conjugation action of Pin(m) and

similarly for the conjugation action of O(n) on SO(n).To solve this problem, we need some background on the representation

theory of a connected compact Lie group G. We use [4, Chapter VI] as ageneral reference. Let T denote a maximal torus in G and let t denote its Liealgebra. Let I∗ ⊆ t∗ denote the set of integral weights on T , i.e., the set oflinear functionals λ : t → R which vanish on the kernel of exp : t → T . There

is a one to one correspondence between I∗ and T given by sending an integralweight λ to the character eλ : T → T defined by eλ(exp(t)) = e2πiλ(t) for allt ∈ t.

Let C denote the closure of a fundamental Weyl chamber C in t∗ and letθ1, . . . , θl ∈ I∗ ∩ C be the corresponding set of positive roots. In particular,

θ1, . . . , θl is a base of t∗ and C = ∑li=1 aiθi : ai ≥ 0. There is a natural order

on C given by λ ≤ η ⇔ η − λ ∈ C. Let W be the Weyl group of G, i.e., thegroup of automorphisms of T induced from inner automorphisms of G. ThenW acts canonically on T , t, t∗ and I∗.

For any finite dimensional complex representation τ of G the equivalenceclass of τ is uniquely determined by its character χτ := tr τ , which is constanton conjugacy classes in G. A virtual character is a linear combination of suchcharacters with integer coefficients. The set R(G) of all virtual characters ofG is called the representation ring of G. It is actually a subring of the ring ofcontinuous functions on G. Every element in R(G) can be written as a (integer)linear combination of irreducible characters, i.e., the characters correspondingto irreducible representations of G. Since the restriction τ |T of a representationτ of G is invariant under conjugation with elements in W (up to equivalence),the restriction of its character χτ to T is conjugation invariant, and hence liesin the set R(T )W of symmetric (i.e., W -invariant) virtual characters of T . By[4, Chapter VI, Proposition (2.1)] the restriction map

res : R(G)→ R(T )W , χ 7→ χ|T

is an isomorphism of rings. Now, for any λ ∈ I∗ we let Wλ = w · λ : w ∈W



denote the W -orbit of λ in I∗ and let

S(λ) =∑

ξ∈Wλ

eξ

denote the symmetrized character corresponding to λ. Combining [4, ChapterVI, Theorem (1.7) and Proposition (2.6)] we get the following version of Weyl’scharacter formula:

Theorem 6.1. For each irreducible representation τ of G there exists a uniquedecomposition

χτ |T = S(λ) +

k∑

i=1

liS(λi)

with pairwise different λ, λ1, . . . , λk ∈ I∗ ∩ K, l1, . . . , lk ∈ Z and λi < λ for all1 ≤ i ≤ k. We call λ ∈ I∗ ∩ C the highest weight of the representation τ . The

map which assigns τ to its highest weight λ induces a bijection between G andI∗ ∩ C.

In what follows we shall denote by χλ ∈ R(G) the character of the irreduciblerepresentation τ with highest weight λ. If γ, λ are weights in I∗ ∩ C, then sois γ + λ and there is a corresponding irreducible character χγ+λ of G. By [4,Chapter VI, (2.8)] we have

Lemma 6.2. For all γ, λ ∈ I∗ ∩ C there is a unique decomposition

χγ · χλ = χγ+λ +∑

µ

lµχµ,

where µ runs through µ ∈ I∗ ∩ C : µ < λ+ γ and 0 ≤ lµ ∈ Z.

A set λ1, . . . , λk of integral weights in I∗ ∩ C is called a fundamentalsystem, if the map

ϕ : Nk0 → I∗ ∩ C, ϕ(l1, . . . , lk) =

k∑

i=1

liλi

is an ordered bijection with respect to the standard order on Nk0 . The cor-responding irreducible representations τ1, . . . , τk are then called a system offundamental representations of G. By [4, Chapter VI (2.10) and (2.11)] wehave

Theorem 6.3. Suppose that G is a connected and simply connected compactLie group. Then there exists a fundamental system λ1, . . . , λk in I∗ ∩ C andthere is a ring isomorphism ψ : Z[X1, . . . , Xk]→ R(G) which sends Xi to χλi .

Combining these results, we get

Lemma 6.4. Let G be a connected and simply connected Lie group and let

λ1, . . . , λk be a fundamental system in I∗ ∩ C. Let γ =∑ki=1 liλi be any

given weight in I∗ ∩ C and let χγ be the corresponding irreducible character



of G. Then there exists a unique polynomial P ∈ Z[X1, . . . , Xk] of order lessthan l := l1 + l2 + . . .+ lk such that

χγ =

k∏

i=1

χliλi + P (χλ1 , . . . , χλk).

Proof. Uniqueness is a direct consequence of Theorem 6.3 above. For existence,we give a proof by induction on the sum l = l1 + l2 + . . . + lk correspondingto γ, which we call the order of γ. If l = 0, then χγ ≡ 1 is the characterof the trivial representation and the formula is true with P = 0 (we use theconvention that the order of the zero-polynomial is −∞). Suppose now thatfor given l > 0 the lemma is true for all m < l. Let γ ∈ I∗ ∩ C with order l,

γ =∑ki=1 liχλi . Without loss of generality we may assume that l1 > 0. By

Lemma 6.2 we have

χγ = χλ1χγ−λ1 −∑

µ<γ

lµχµ

for suitable lµ ∈ N0. Since µ < γ, the order of µ is less than the order of γ.Thus, by the induction hypothesis, there exists a polynomial Pµ with order < lsuch that χµ = Pµ(χλ1 , . . . , χλk). Similarly, the induction hypothesis gives adecomposition

χγ−λ1 = χl1−1λ1

k∏

i=2

χliλi + Pγ−λ1(χλ1 , . . . , χλk),

such that the order of Pγ−λ is smaller than l− 1. The result then follows withP = X1Pγ−λ1 −

∑µ<γ lµPµ.

We are now coming back to the special case of the group G = Spin(n) withn = 2m and m ≥ 2. This group is simply connected and connected and by [4,Chapter VI, Theorem (6.2)] a system of fundamental representations is givenby the representations

Λ1, . . . ,Λm−2,Σ+,Σ−

defined as follows: The representations Λi act on the complexification Λi(Cn)of the ith exterior power Λi(Rn) by inflating the canonical action of SO(n)on Λi(Rn) to Spin(n). Note that these representations extend canonically toPin(n) (resp. to O(n), if we view them as representations of SO(n)), whichimplies that the Λi are stable (up to equivalence) under conjugation by ele-ments in Pin(n) (resp. O(n)). It is also clear that the Λi are non-negative, i.e,Λi(−1) = 1.

The representations Σ+,Σ− are the half-spin representations on the spacesS+, S− defined as follows: By the isomorphism Cl(n) ∼= M2m(C) we find acanonical irreducible action of the complex Clifford algebra Cl(n) on S := C2m .

Since J2 = (−1)m, for J = e1 · · · en, it follows that J := imJ satisfies J = J∗ =

J−1, which implies that S decomposes into the direct sum of two orthogonaleigenspaces S+, S− for the eigenvalues ±1 of J . Since JxJ = JxJ∗ = x forall x ∈ Cl(n)0, these spaces are invariant under the action of Cl(n)0, and then



restrict to unitary representations Σ± of Spin(n) ⊆ Cl(n)0. One easily checksthat conjugation by e1 ∈ Pin(n) \ Spin(n) intertwines these representations.

We therefore see that Σ+,Σ− forms one orbit of length two in Spin(n) underconjugation by Pin(n). By construction, the representations Σ± are negativerepresentations, i.e., Σ±(−1) = −1.

We are now ready to prove the following proposition, which will give thelast step in the proof of Theorem 5.5 of the previous section.

Proposition 6.5. Let n = 2m ≥ 4. Then the following are true:

(i) If x ∈ Pin(n) \ Spin(n) and τ is a negative irreducible representationof Spin(n), then τ 6∼= x · τ . Thus, for the orbit sets O1 and O2 in

Spin(n)−

we get O1 = ∅ and O2 is countably infinite.

(ii) For the action of O(n) on SO(n) both orbit sets O1 and O2 are count-ably infinite.

Proof. Let χ1, . . . , χm−2, χ+, χ− denote the characters corresponding to thefundamental representations Λ1, . . . ,Λm−2,Σ±. Let τ be any negative irre-ducible representation of Spin(n) with character χτ . By Theorem 6.3 thereexists a unique Polynomial Q ∈ Z[X1, . . . , Xm−2, X+, X−] such that χτ =Q(χ1, . . . , χm+2, χ+, χ−). By Lemma 6.4 the polynomial Q is of the form

(m−2∏

i=1

X lii

)Xl++ X

l−− + P (X1, . . . , Xm−2, X+, X−)

with the order of P less than l = l1 + · · ·+ lm−2 + l+ + l−. Since τ is negative,we have χτ (−x) = −χτ (x) for all gx ∈ Spin(n). Since for all x ∈ Spin(n) wehave χi(−x) = χi(x), for all 1 ≤ i ≤ m− 2, and χ±(−x) = −χ±(x) we get

χτ = −(−1)l++l−

(m−2∏

i=1

χlii

)χl++ χ

l−− + P (χ1, . . . , χn−2, χ+, χ−)

with the order of P less than l. By the uniqueness of the polynomial represen-tation of χτ it follows that (−1)l++l− = −1 and, in particular, that l+ 6= l−.

Suppose now that x ∈ Pin(n)\Spin(n). Since x·χi = χi for all 1 ≤ i ≤ m−2and xχ+ = χ− (and vice versa) we get

xχτ = Q(xχ1, . . . , xχm−2, xχ+, xχ−)

=

(m−2∏

i=1

(xχi)li

)(xχ+)l+(xχ−)l− + P (xχ1, . . . , xχm−2, xχ+, xχ−)

=

(m−2∏

i=1

χlii

)χl+− χ

l−+ + P (χ1, . . . , χm−2, χ−, χ+)

= Q(χ1, . . . , χm−2, χ+, χ−).

Since l+ 6= l− we have Q 6= Q, hence χxτ = xχτ 6= χτ , and therefore xτ 6∼= τ .This proves (i).



For the proof of (ii) note first that SO(n) = Spin(n)+, the set of irreducible

representations τ of Spin(n) with τ(−1) = 1. Writing its character χτ as

Q(χ1, . . . , χm+2, χ+, χ−) as above, we see that τ ∈ SO(n) if and only if l++l− iseven. Then a similar computation as above shows that for x ∈ Pin(n)\Spin(n)we get

xτ ∼= τ ⇔ xχτ = χτ ⇔ l+ = l−.

It is now clear that there are infinitely many representations which are fixedby conjugation and there are also infinitely many pairs of conjugate represen-

tations in SO(n).

Proof of Theorem 5.5. The proof now follows from the above proposition to-gether with the discussion of the even case preceding Proposition 5.4.

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[8] M. Karoubi, Equivariant K-theory of real vector spaces and real projective spaces, Topol-ogy Appl. 122 (2002), no. 3, 531–546. MR1911698 (2003h:19008)

[9] G. G. Kasparov, The operator K-functor and extensions of C∗-algebras, Izv. Akad. NaukSSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719. MR0582160 (81m:58075)

[10] F. Malow. Das Deformationsbild der Baum-Connes-Vermutung fur fast zusam-menhangende Lie-Gruppen. Ph.D thesis, Munster 2007.

Received March 11, 2009; accepted July 8, 2009

Siegfried Echterhoff and Oliver PfanteWestfalische Wilhelms-Universitat Munster, Mathematisches InstitutEinsteinstr. 62, D-48149 Munster, GermanyE-mail: [email protected]

URL: http://wwwmath.uni-muenster.de/u/echters/E-mail: [email protected]




Ocneanu cells and Boltzmann weights for the

SU(3) ADE graphs

David E. Evans and Mathew Pugh



Abstract. We determine the cells, whose existence has been announced by Ocneanu, on

all the candidate nimrep graphs except E(12)4 proposed by di Francesco and Zuber for the

SU(3) modular invariants classified by Gannon. This enables the Boltzmann weights to becomputed for the corresponding integrable statistical mechanical models and provide theframework for studying corresponding braided subfactors to realize all the SU(3) modularinvariants as well as a framework for a new SU(3) planar algebra theory.

1. Introduction

In the last twenty years, a very fruitful circle of ideas has developed link-ing the theory of subfactors with modular invariants in conformal field theory.Subfactors have been studied through their paragroups, planar algebras andhave serious contact with free probability theory. The understanding and clas-sification of modular invariants is significant for conformal field theory andtheir underlying statistical mechanical models. These areas are linked throughthe use of braided subfactors and α-induction which in particular for SU(2)subfactors and SU(2) modular invariants invokes ADE classifications on bothsides. This paper is the first of our series to study more precisely these con-nections in the context of SU(3) subfactors and SU(3) modular invariants.The aim is to understand them not only through braided subfactors and α-induction but introduce and develop a pertinent planar algebra theory and freeprobability.

A group acting on a factor can be recovered from the inclusion of its fixedpoint algebra. A general subfactor encodes a more sophisticated symmetry ora way of handling non group like symmetries including but going beyond quan-tum groups [18]. The classification of subfactors was initiated by Jones [29]

96 David E. Evans and Mathew Pugh

who found that the minimal symmetry to understand the inclusion is throughthe Temperley-Lieb algebra. This arises from the representation theory ofSU(2) or dually certain representations of Hecke algebras. All SU(2) modularinvariant partition functions were classified by Cappelli, Itzykson and Zuber[11, 12] using ADE Coxeter-Dynkin diagrams and their realization by braidedsubfactors is reviewed and referenced in [15]. There are a number of invari-ants (encoding the symmetry) one can assign to a subfactor, and under certaincircumstances they are complete at least for hyperfinite subfactors. Popa [39]axiomatized the inclusions of relative commutants in the Jones tower, andJones [30] showed that this was equivalent to his planar algebra description.Here one is naturally forced to work with nonamenable factors through freeprobabilistic constructions e.g. [27]. In another vein, Banica and Bisch [3]understood the principal graphs, which encode only the multiplicities in theinclusions of the relative commutants, and more generally nimrep graphs interms of spectral measures, and so provide another way of understanding thesubfactor invariants.

In our series of papers we will look at this in the context of SU(3), throughthe subfactor theory and their modular invariants, beginning here and contin-uing in [20, 21, 22, 23, 24]. The SU(3) modular invariants were classified byGannon [26]. Ocneanu [37] announced that all these modular invariants wererealized by subfactors, and most of these are understood in the literature andwill be reviewed in the sequel [20]. A braided subfactor automatically givesa modular invariant through α-induction. This α-induction yields a represen-tation of the Verlinde algebra or a nimrep - which yields multiplicity graphsassociated to the modular invariants (or at least associated to the inclusion,as a modular invariant may be represented by wildly differing inclusions andso may possess inequivalent but isospectral nimreps, as is the case for E(12)).In the case of the SU(3) modular invariants, candidates of these graphs wereproposed by di Francesco and Zuber [14] by looking for graphs whose spectrumreproduced the diagonal part of the modular invariant, aided to some degreeby first listing the graphs and spectra of fusion graphs of the finite subgroupsof SU(3). In the SU(2) situation there is a precise relation between the ADECoxeter-Dynkin graphs and finite subgroups of SU(2) as part of the McKaycorrespondence. However, for SU(3), the relation between nimrep graphs andfinite subgroups of SU(3) is imprecise and not a perfect match. For SU(2), anaffine Dynkin diagram describing the McKay graph of a finite subgroup givesrise to a Dynkin diagram describing a nimrep or the diagonal part of a modularinvariant by removing the vertex corresponding to the identity representation.Di Francesco and Zuber found graphs whose spectrum described the diagonalpart of a modular invariant by taking the list of McKay graphs of finite sub-groups of SU(3) and removing vertices. Not every modular invariant couldbe described in this way, and not every finite subgroup yielded a nimrep for amodular invariant. In higher rank SU(N), the number of finite subgroups willincrease but the number of exceptional modular invariants should decrease, so


Ocneanu cells and Boltzmann weights for SU(3) ADE graphs 97

this procedure is even less likely to be accurate. Evans and Gannon have sug-gested an alternative way of associating finite subgroups to modular invariants,by considering the largest finite stabilizer groups [16].

A modular invariant which is realized by a subfactor will yield a graph. Toconstruct these subfactors we will need some input graphs which will actu-ally coincide with the output nimrep graphs - SU(3) ADE graphs. The aimof this series of papers is to study the SU(3) ADE graphs, which appear inthe classification of modular invariant partition functions from numerous view-points including the determination of their Boltzmann weights in this paper,representations of SU(3)-Temperley-Lieb or Hecke algebra [20], a new notionof SU(3)-planar algebras [21] and their modules [22], and spectral measures[23, 24].

As pointed out to us by Jean-Bernard Zuber, there is a renewal of interest(by physicists) in these SU(3) and related theories, in connection with topo-logical quantum computing [1] and by Joost Slingerland in connection withcondensed matter physics [2] where we see that α-induction is playing a keyrole.

We begin however in this paper by computing the numerical values of theOcneanu cells, and consequently representations of the Hecke algebra, for theADE graphs. These cells give numerical weight to Kuperberg’s [32] diagramof trivalent vertices—corresponding to the fact that the trivial representationis contained in the triple product of the fundamental representation of SU(3)through the determinant. They will yield in a natural way, representations ofan SU(3)-Temperley-Lieb or Hecke algebra. (For SU(2) or bipartite graphs,the corresponding weights (associated to the diagrams of cups or caps), arisein a more straightforward fashion from a Perron-Frobenius eigenvector, givinga natural representation of the Temperley-Lieb algebra or Hecke algebra). We

have been unable thus far to compute the cells for the exceptional graph E(12)4 .

This graph is meant to be the nimrep for the modular invariant conjugate to

the Moore-Seiberg invariant E(12)MS [33]. However we will still be able to realize

this modular invariant by subfactors in [20] using [19]. For the orbifold graphs

D(3k), k = 2, 3, . . . , orbifold conjugate D(n)∗, n = 6, 7, . . . , and E(12)1 we

compute solutions which satisfy some additional condition, but for the othergraphs we compute all the Ocneanu cells, up to equivalence. The existenceof these cells has been announced by Ocneanu (e.g. [36, 37]), although thenumerical values have remained unpublished. Some of the representations ofthe Hecke algebra have appeared in the literature and we compare our results.

For the A graphs, our solution for the Ocneanu cells W gives an identicalrepresentation of the Hecke algebra to that of Jimbo et al. [28] given in (21).Our cells for the A(n)∗ graphs give equivalent Boltzmann weights to thosegiven by Behrend and Evans in [4]. In [14], di Francesco and Zuber give arepresentation of the Hecke algebra for the graphs D(6)∗ and E(8), whilst in

[41] a representation of the Hecke algebra is computed for the graphs E(12)1 and

E(24). Our solutions for the cells W give an identical Hecke representation for



E(8) and an equivalent Hecke representation for E(12)1 . However, for E(24), our

cells give inequivalent Boltzmann weights. In [25], Fendley gives Boltzmannweights for D(6) which are not equivalent to those we obtain, but which areequivalent if we take one of the weights in [25] to be the complex conjugate ofwhat is given.

Subsequently, we will use these weights, their existence and occasionallymore precise use of their numerical values. Here we outline some of the flavorof these applications. We use these cells to define an SU(3) analogue of theGoodman-de la Harpe Jones construction of a subfactor, where we embedthe SU(3)-Temperley-Lieb or Hecke algebra in an AF path algebra of theSU(3) ADE graphs. Using this construction, we realize all the SU(3) modularinvariants by subfactors [20].

We will then [21, 22] look at the SU(3)-Temperley-Lieb algebra and theSU(3)-GHJ subfactors from the viewpoint of planar algebras. We give a di-agrammatic representation of the SU(3)-Temperley-Lieb algebra, and showthat it is isomorphic to Wenzl’s representation of a Hecke algebra. Generaliz-ing Jones’s notion of a planar algebra, we construct an SU(3)-planar algebrawhich will capture the structure contained in the SU(3) ADE subfactors. Weshow that the subfactor for an ADE graph with a flat connection has a de-scription as a flat SU(3)-planar algebra. We introduce the notion of modulesover an SU(3)-planar algebra, and describe certain irreducible Hilbert SU(3)-TL-modules. A partial decomposition of the SU(3)-planar algebras for theADE graphs is achieved. Moreover, in [23, 24] we consider spectral measuresfor the ADE graphs in terms of probability measures on the circle T. We gen-eralize this to SU(3), and in particular obtain spectral measures for the SU(3)graphs. We also compare various Hilbert series of dimensions associated toADE models for SU(2), and compute the Hilbert series of certain q-deformedCalabi-Yau algebras of dimension 3.

In Section 2, we specify the graphs we are interested in, and in Section3 recall the notion of cells due to Ocneanu which we will then compute inSections 4 - 14.

2. ADE Graphs

We enumerate the graphs we are interested in. These will eventually providethe nimrep classification graphs for the list of SU(3) modular invariants, but atthis point, they will only provide a framework for some statistical mechanicalmodels with configurations spaces built from these graphs together with someBoltzmann weights which we will need to construct. However, for the sake ofclarity of notation, we start by listing the SU(3) modular invariants. Thereare four infinite series of SU(3) modular invariants: the diagonal invariants,labelled by A, the orbifold invariants D, the conjugate invariants A∗, and theorbifold conjugate invariants D∗. These will provide four infinite families ofgraphs, written as A, the orbifold graphs D, the conjugate graphs A∗, and theorbifold conjugate graphs D∗, shown in Figures 4, 7, 10, 11 and 12. There



are also exceptional SU(3) modular invariants, i.e. invariants which are notdiagonal, orbifold, or their conjugates, and there are only finitely many of

these. These are E(8) and its conjugate, E(12), E(12)MS and its conjugate, and

E(24). The exceptional invariants E(12) and E(24) are self-conjugate.The modular invariants arising from SU(3)k conformal embeddings are:

• D(6): SU(3)3 ⊂ SO(8)1, also realized as an orbifold SU(3)3/Z3,• E(8): SU(3)5 ⊂ SU(6)1, plus its conjugate,• E(12): SU(3)9 ⊂ (E6)1,

• E(12)MS : Moore-Seiberg invariant, an automorphism of the orbifold in-

variant D(12) = SU(3)9/Z3, plus its conjugate,• E(24): SU(3)21 ⊂ (E7)1.

These modular invariants will be associated with graphs, as follows. Therewill be one graph E(8) for the E(8) modular invariant and its orbifold graph

E(8)∗ for its conjugate invariant as in Figure 13. The modular invariants E(12)MS

and its conjugate will be associated to the graphs E(12)5 and E(12)

4 respectively

as in Figure 15. The exceptional invariant E(12) is self-conjugate but has as-

sociated to it two isospectral graphs E(12)1 and E(12)

2 as in Figure 14. The

invariant E(24) is also self-conjugate and has associated to it one graph E(24)

as in Figure 16. The modular invariants themselves play no role in this pa-per other than to help label these graphs. In the sequel to this paper [20] wewill use the Boltzmann weights obtained here to construct braided subfactors,which via α-induction [5, 6, 7, 8, 9, 10] will realize the corresponding modularinvariants. Furthermore, α-induction naturally provides a nimrep or repre-sentation of the original fusion rules or Verlinde algebra. The correspondingnimreps will then be computed and we will recover the original input graph.The theory of α-induction will guarantee that the spectra of these graphs aredescribed by the diagonal part of the corresponding modular invariant. Thusdetailed information about the spectra of these graphs will naturally followfrom this procedure and does not need to be computed at this stage. Many ofthese modular invariants are already realized in the literature and this will bereviewed in the sequel to this paper [20].

3. Ocneanu Cells

Let Γ be SU(3) and Γ its irreducible representations. One can associateto Γ a McKay graph GΓ whose vertices are labelled by the irreducible repre-

sentations of Γ, where for any pair of vertices i, j ∈ Γ the number of edgesfrom i to j are given by the multiplicity of j in the decomposition of i ⊗ ρinto irreducible representations, where ρ is the fundamental irreducible rep-resentation of SU(3), and which, along with its conjugate representation ρ,

generates Γ. The graph GΓ is made of triangles, corresponding to the factthat the fundamental representation ρ satisfies ρ⊗ ρ⊗ ρ ∋ 1. We define mapss, r from the edges of GΓ to its vertices, where for an edge γ, s(γ) denotesthe source vertex of γ and r(γ) its range vertex. For the graph GΓ, a triangle



(αβγ)ijk = i

α- j

β- k

γ- i is a closed path of length 3 consisting of edges

α, β, γ of GΓ such that s(α) = r(γ) = i, s(β) = r(α) = j and s(γ) = r(β) = k.

For each triangle (αβγ)ijk , the maps α, β and γ are composed:

iid⊗det∗

- i⊗ ρ⊗ ρ⊗ ρ γ⊗id- k ⊗ ρ⊗ ρ β⊗id

- j ⊗ ρ α⊗id- i,

and since i is irreducible, the composition is a scalar. Then for every suchtriangle on GΓ there is a complex number, called an Ocneanu cell. There isa gauge freedom on the cells, which comes from a unitary change of basis inHom[i⊗ ρ, j] for every pair i, j.

These cells are axiomatized in the context of an arbitrary graph G whoseadjacency matrix has Perron-Frobenius eigenvalue [3] = [3]q, although in prac-tice it will be one of the ADE graphs. Note however we do not require Gto be three-colorable (e.g. the graphs A∗ which will be associated to theconjugate modular invariant). Here the quantum number [m]q be defined as[m]q = (qm− q−m)/(q− q−1). We will frequently denote the quantum number[m]q simply by [m], for m ∈ N. Now [3]q = q2 + 1 + q−2, so that q is easilydetermined from the eigenvalue of G. The quantum number [2] = [2]q is thensimply q + q−1. If G is an ADE graph, the Coxeter number n of G is thenumber in parentheses in the notation for the graph G, e.g. the exceptionalgraph E(8) has Coxeter number 8, and q = eπi/n. With this q, the quantumnumbers [m]q satisfy the fusion rules for the irreducible representations of thequantum group SU(2)n, i.e.

(1) [a]q [b]q =∑

c

[c]q,

where the summation is over all integers |b − a| ≤ c ≤ min(a + b, 2n− a − b)such that a+ b+ c is even.

We define a type I frame in an arbitrary G to be a pair of edges α, α′ whichhave the same start and endpoint. A type II frame will be given by four edgesαi, i = 1, 2, 3, 4, such that s(α1) = s(α4), s(α2) = s(α3), r(α1) = r(α2) andr(α3) = r(α4).

Definition 3.1 ([37]). Let G be an arbitrary graph with Perron-Frobeniuseigenvalue [3] and Perron-Frobenius eigenvector (φi). A cell system W on

G is a map that associates to each oriented triangle (αβγ)ijk in G a complex

number W((αβγ)ijk

)with the following properties:

(i) for any type I frame in G we have

(2)



(ii) for any type II frame in G we have

(3)

In [32], Kuperberg defined the notion of a spider—a way of depicting theoperations of the representation theory of groups and other group-like objectswith certain planar graphs, called webs (hence the term “spider”). Certainspiders were defined in terms of generators and relations, isomorphic to therepresentation theories of rank two Lie algebras and the quantum deformationsof these representation theories. This formulation generalized a well-knownconstruction for A1 = su(2) by Kauffman [31]. For the A2 = su(3) case, theA2 webs are illustrated in Figure 1.

Figure 1. A2 webs

The A2 web space generated by these A2 webs satisfy the Kuperberg rela-tions, which are relations on local parts of the diagrams:

K1:

K2:

K3:

The rules (2), (3) correspond precisely to evaluating the Kuperberg relationsK2, K3 respectively, associating a cell W (α,β,γ) to an incoming trivalent

vertex, and W (α,β,γ) to an outgoing trivalent vertex, as in Figure 2.

Figure 2. Cells associated to trivalent vertices



We define the connection

Xρ1,ρ2ρ3,ρ4 =

lρ1- i

k

ρ3 ? ρ4- j

ρ2?

for G by

(4) Xρ1,ρ2ρ3,ρ4 = q

23 δρ1,ρ3δρ2,ρ4 − q−

13Uρ1,ρ2ρ3,ρ4 ,

where Uρ1,ρ2ρ3,ρ4 is given by the representation of the Hecke algebra, and is definedby

(5) Uρ1,ρ2ρ3,ρ4 =∑

λ

φ−1s(ρ1)φ

−1r(ρ2)

W ((λ,ρ3,ρ4)j,l,k )W ((λ,ρ1,ρ2)

j,l,i ).

This definition of the connection is really Kuperberg’s braiding of [32].The above connection corresponds to the natural braid generator gi, which

is the Boltzmann weight at criticality, and which satisfy

gigj = gjgi if |j − i| > 1,(6)

gigi+1gi = gi+1gigi+1.(7)

It was claimed in [36] that the connection satisfies the unitarity property ofconnections

(8)∑

ρ3,ρ4

Xρ1,ρ2ρ3,ρ4 X

ρ′1,ρ′2

ρ3,ρ4 = δρ1,ρ′1δρ2,ρ′2 ,

and the Yang-Baxter equation

(9)∑

σ1,σ2,σ3

Xσ1,σ2ρ1,ρ2 Xρ3,ρ4

σ1,σ3Xσ3,ρ5σ2,ρ6 =

∑

σ1,σ2,σ3

Xρ3,σ2ρ1,σ1

Xσ1,σ3ρ2,ρ6 Xρ4,ρ5

σ2,σ3.

The Yang-Baxter equation (9) is represented graphically in Figure 3. We givea proof that the connection (4) satisfies these two properties.

Figure 3. The Yang-Baxter equation

Lemma 3.2. If the conditions in Definition 3.1 are satisfied, the connectiondefined in (4) satisfies the unitarity property (8) and the Yang-Baxter equation(9).



Proof. We first show unitarity.

∑

ρ3,ρ4

Xρ1,ρ2ρ3,ρ4X

ρ′1,ρ′2

ρ3,ρ4

=∑

ρ3,ρ4

(q

23 δρ1,ρ3δρ2,ρ4 − q−

13

∑

λ

1

φs(ρ1)φr(ρ2)Wρ3,ρ4,λWρ1,ρ2,λ

)

×(q−

23 δρ′1,ρ3δρ′2,ρ4 − q

13

∑

λ

1

φs(ρ1)φr(ρ2)Wρ′1,ρ

′2,λWρ3,ρ4,λ

)

= δρ1,ρ′1δρ3,ρ′3 +∑

ρ3,ρ4λ,λ′

1

φ2s(ρ1)φ

2r(ρ2)

Wρ3,ρ4,λWρ1,ρ2,λWρ′1,ρ′2,λWρ3,ρ4,λ

−∑

ρ3,ρ4,λ

1

φs(ρ1)φr(ρ2)

(qδρ1,ρ3δρ2,ρ4Wρ′1,ρ

′2,λWρ3,ρ4,λ

+q−1δρ′1,ρ3δρ′2,ρ4Wρ3,ρ4,λWρ1,ρ2,λ

)

= δρ1,ρ′1δρ3,ρ′3 +∑

λ,λ′

1

φ2s(ρ1)φ

2r(ρ2)

Wρ1,ρ2,λWρ′1,ρ′2,λ

[2]φs(ρ3)φr(ρ4)δλ,λ′

− (q + q−1)∑

λ

1

φs(ρ1)φr(ρ2)Wρ′1,ρ

′2,λWρ1,ρ2,λ

= δρ1,ρ′1δρ3,ρ′3 ,

since q + q−1 = [2], where we have used Ocneanu’s type I equation (3) in thepenultimate equality.

We now show that the connection satisfies the Yang-Baxter equation. Forthe left hand side of (9) we have

∑

σ1,σ2,σ3

Xσ1,σ2ρ1,ρ2 X

ρ3,ρ4σ1,σ3

Xσ3,ρ5σ2,ρ6

=∑

σ1,σ2,σ3

(q

23 δρ1,σ1δρ2,σ2 − q−

13Uσ1,σ2

ρ1,ρ1

)(q−

23 δσ1,ρ3δσ3,ρ4 − q

13Uρ3,ρ4σ1,σ3

)

×(q−

23 δσ2,σ3δρ6,ρ5 − q

13Uσ3,ρ5

σ2,ρ6

)

= q2δρ1,ρ3δρ2,ρ4δρ5,ρ6 − qδρ1,ρ3 Uρ4,ρ5ρ2,ρ6 − qδρ5,ρ6 Uρ3,ρ4ρ1,ρ2 − qδρ5,ρ6 Uρ3,ρ4ρ1,ρ2

+∑

σ3

Uρ3,ρ4ρ1,σ3Uσ3,ρ5ρ2,ρ6 +

∑

σ2

Uρ3,σ2ρ1,ρ2 Uρ4,ρ5σ2,ρ6 + δρ5,ρ6

∑

σ1,σ2

Uσ1,σ2ρ1,ρ2 Uρ3,ρ4σ1,σ2

− q−1∑

σi

Uσ1,σ2ρ1,ρ2 Uρ3,ρ4σ1,σ3

Uσ3,ρ5σ2,ρ6

= q2δρ1,ρ3δρ2,ρ4δρ5,ρ6 − qδρ1,ρ3 Uρ4,ρ5ρ2,ρ6 − 2qδρ5,ρ6 Uρ3,ρ4ρ1,ρ2

+∑

σ3


∑

σ2

Uρ3,σ2ρ1,ρ2 Uρ4,ρ5σ2,ρ6



+ δρ5,ρ6∑

σ1,σ2λ,λ′

1

φs(ρ1)φr(ρ2)φs(ρ3)φr(ρ4)Wρ1,ρ2,λWσ1,σ2,λWσ1,σ2,λ′Wρ3,ρ4,λ′

− q−1∑

σi,λ

λ′,λ′′

1

φ2s(ρ1)φr(ρ2)φr(ρ4)φs(σ2)φr(ρ6)

Wρ1,ρ2,λWρ3,ρ4,λ′

×Wσ1,σ2,λWσ1,σ3,λ′Wσ2,ρ6,λ′′Wσ1,ρ5,λ′′

= q2δρ1,ρ3δρ2,ρ4δρ5,ρ6 − qδρ1,ρ3 Uρ4,ρ5ρ2,ρ6 − 2qδρ5,ρ6 Uρ3,ρ4ρ1,ρ2

+∑

σ3


∑

σ2

Uρ3,σ2ρ1,ρ2 Uρ4,ρ5σ2,ρ6

+ δρ5,ρ6∑

λ,λ′

1

φs(ρ1)φr(ρ2)φs(ρ3)φr(ρ4)Wρ1,ρ2,λWρ3,ρ4,λ′ [2]φr(ρ2)φs(ρ1)δλ,λ′

− q−1∑

λ,λ′

1

φ2s(ρ1)φr(ρ2)φr(ρ4)φr(ρ6)

Wρ1,ρ2,λWρ3,ρ4,λ′

×(δλ,ρ6δλ′,ρ5φr(ρ2)φr(ρ6)φr(ρ4) + δλ,λ′δρ5,ρ6φs(ρ1)φr(ρ2)φr(ρ6)

)

= q2δρ1,ρ3δρ2,ρ4δρ5,ρ6 − qδρ1,ρ3 Uρ4,ρ5ρ2,ρ6 − qδρ5,ρ6 Uρ3,ρ4ρ1,ρ2 +∑

σ3

Uρ3,ρ4ρ1,σ3Uσ3,ρ5ρ2,ρ6

+∑

σ2

Uρ3,σ2ρ1,ρ2 Uρ4,ρ5σ2,ρ6 − q−1 1

φs(ρ1)Wρ1,ρ2,ρ6Wρ3,ρ4,ρ5 .

Computing the right hand side of (9) in the same way, we arrive at the sameexpression.

4. Computation of the cells W for ADE graphs

In the remaining sections we will compute cells systems W for each ADEgraph G, with the exception of the graph E(12)

4 .

Let (α,β,γ)i,j,k be the triangle i

α- j

β- k

γ- i in G. For most of the

ADE graph, using the equations (2) and (3) only, we can compute the cells

up to choice of phase W ((α,β,γ)i,j,k ) = λα,β,γi,j,k |W ((α,β,γ)

i,j,k )| for some λα,β,γi,j,k ∈ T,

and also obtain some restrictions on the values which the phases λα,β,γi,j,k may

take. However, for the graph D(n)∗, n = 5, 6, . . . , we impose a Z3 symmetry

on our solutions, whilst for the graphs D(3k), k = 2, 3, . . . , and E(12)1 we seek

an orbifold solution obtained using the identification of the graphs D(3k), E(12)1

as Z3 orbifolds of A(3k), E(12)2 respectively. There is still much freedom in the

actual choice of phases, so that the cell system is not unique. We thereforedefine an equivalence relation between two cell systems:

Definition 4.1. Two families of cells W1, W2 which give a cell system forG are equivalent if, for each pair of adjacent vertices i, j of G, we can find afamily of unitary matrices (u(σ1, σ2))σ1,σ2 , where σ1, σ2 are any pair of edges



from i to j, such that

(10) W1((σ,ρ,γ)i,j,k ) =

∑

σ′,ρ′,γ′

u(σ, σ′)u(ρ, ρ′)u(γ, γ′)W2((σ′,ρ′,γ′)i,j,k ),

where the sum is over all edges σ′ from i to j, ρ′ from j to k, and γ′ from k toi.

Lemma 4.2. Let W1, W2 be two equivalent families of cells, and X(1), X(2)

the corresponding connections defined using cells W1, W2 respectively. ThenX(1) and X(2) are equivalent in the sense of [18, p.542], i.e. there exists a setof unitary matrices (u(ρ, σ))ρ,σ such that

X(1)ρ1,ρ2ρ3,ρ4 =

∑

σi

u(ρ3, σ3)u(ρ4, σ4)u(ρ1, σ1)u(ρ2, σ2)X(2)σ1,σ2σ3,σ4

.

Let Wl((σ,ρ,γ)i,j,k ) = λ

(l)σ,ρ,γi,j,k |Wl((σ,ρ,γ)

i,j,k )|, for l = 1, 2, be two families of

cells which give cell systems. If |W1((σ,ρ,γ)i,j,k )| = |W2((σ,ρ,γ)

i,j,k )|, so that W1

and W2 differ only up to phase choice, then the equation (10) becomes

(11) λ(1)σ,ρ,γi,j,k =

∑

σ′,ρ′,γ′

u(σ, σ′)u(ρ, ρ′)u(γ, γ′)λ(2)σ,ρ,γi,j,k .

For graphs with no multiple edges we write i,j,k for the triangle (α,β,γ)i,j,k .

For such graphs, two solutions W1 and W2 differ only up to phase choice, and(11) becomes

(12) λ(1)i,j,k = uσuρuγλ

(2)i,j,k,

where uσ, uρ, uγ ∈ T and σ is the edge from i to j, ρ the edge from j to k andγ the edge from k to i.

We will write U (x,y) for the matrix indexed by the vertices of G, with entriesgiven by Uρ1,ρ2ρ3,ρ4 for all edges ρi, i = 1, 2, 3, 4 on G such that s(ρ1) = s(ρ3) = x,

r(ρ2) = r(ρ4) = y, i.e. [U (s(ρ1),r(ρ2))]r(ρ1),r(ρ3) = Uρ1,ρ2ρ3,ρ4 .We first present some relations that the quantum numbers [a]q satisfy, which

are easily checked:

Lemma 4.3.

(i) If q = exp(iπ/n) then [a]q = [n− a]q, for any a = 1, 2, . . . , n− 1,(ii) For any q, [a]q − [a− 2]q = [2a− 2]q/[a− 1]q, for any a ∈ N,(iii) For any q, [a]2q−[a−1]q[a+1]q = 1 and [a]q[a+b]q−[a−1]q[a+b+1]q =

[b+ 1]q, for any a ∈ N.

5. A graphs

The infinite graph A(∞) is illustrated in Figure 4, whilst for finite n, thegraphs A(n) are the subgraphs of A(∞), given by all the vertices (λ1, λ2) suchthat λ1 + λ2 ≤ n − 3, and all the edges in A(∞) which connect these ver-tices. The apex vertex (0, 0) is the distinguished vertex. For the triangle(i1,j1)(i2,j2)(i3,j3) = (i1, j1) - (i2, j2) - (i3, j3) - (i1, j1) in A(n) we



Figure 4. The infinite graph A(∞)

will use the notation W(i,j) for the cell W ((i,j)(i+1,j)(i,j+1)) and W∇(i,j) forthe cell W ((i+1,j)(i,j+1)(i+1,j+1)).

Theorem 5.1. There is up to equivalence a unique set of cells for A(n), n <∞, given by

W(k,m) =√

[k + 1][k + 2][m+ 1][m+ 2][k +m+ 1][k +m+ 2]/[2],(13)

W∇(k,m) =√

[k + 1][k + 2][m+ 1][m+ 2][k +m+ 2][k +m+ 3]/[2],(14)

for all k,m ≥ 0. For the graph A(∞) with Perron-Frobenius eigenvalue α ≥ 3,there is a solution given by replacing [j] by [j]q where q = ex for any x ∈ Rsuch that α = [3]q.

Proof. Let n <∞. We first prove the equalities

|W(k,m)| =√

[k + 1][k + 2][m+ 1][m+ 2][k +m+ 1][k +m+ 2]/[2],(15)

|W∇(k,m)| =√

[k + 1][k + 2][m+ 1][m+ 2][k +m+ 2][k +m+ 3]/[2],(16)

by induction on k,m. The Perron-Frobenius eigenvector for A(n) is [13]:

(17) φλ =[λ1 + 1]q[λ2 + 1]q[λ1 + λ2 + 2]q

[2].

For the type I frame(0,0)• -

(1,0)• equation (2) gives |W(0,0)|2 = [2][3],

whilst from the type I frame(1,0)• -

(0,1)• we obtain |W(0,0)|2+|W∇(0,0)|2 =

[2][3]2, giving |W∇(0,0)|2 = [3][4]. We assume (15) and (16) are true for (k,m) =(p, q). We first show (15) is true for (k,m) = (p+ 1, q) and (k,m) = (p, q + 1)

(see Figure 5). From the type I frame(p+1,q+1)• -

(p+1,q)• we get

|W(p+1,q)|2 + |W∇(p,q)|2 = [p+ 2]2[q + 1][q + 2][p+ q + 2][p+ q + 3]/[2],



and substituting in for |W∇(p,q)|2 we obtain

|W(p+1,q)|2

= [p+ 2][q + 1][q + 2][p+ q + 2][p+ q + 3]([2][p+ 2]− [p+ 1])/[2]2

= [p+ 2][p+ 2][q + 1][q + 2][p+ q + 2][p+ q + 3]/[2]2.

Similarly, from the type I frame(p,q+1)• -

(p+1,q+1)• we get

|W(p,q+1)|2 = [p+ 1][p+ 2][q + 2][q + 3][p+ q + 2][p+ q + 3]/[2]2,

as required.

Figure 5. Triangles in A(n)

For k,m ≥ 0, the equality in (16) follows from (15) by considering the type

I frame(k+1,m)• -

(k,m+1)• . We get

|W(k,m)|2 + |W∇(k,m)|2 = [k + 1][k + 2][m+ 1][m+ 2][k +m+ 2]2/[2],

and substituting in for |W(k,m)|2 we obtain

|W∇(k,m)|2 = [k + 1][k + 2][m+ 1][m+ 2][k +m+ 2]

× ([2][k +m+ 2]− [k +m+ 1])/[2]2

= [k + 1][k + 2][m+ 1][m+ 2][k +m+ 2][k +m+ 3]/[2]2.

Hence (15) and (16) are true for all k,m ≥ 0.There is no restriction on the choice of phase for A(n), so any choice is a so-

lution. We now turn to the uniqueness of these cells. Let W ♯ be another family

of cells, with W ♯(k,m) = λ(k,m)|W(k,m)| and W ♯

∇(k,m) = λ′(k,m)|W∇(k,m)| (any

other solution must be of this form since there are no double edges on A(n)).

We label the edges of A(n) by σ(j)i , ρ

(j)i , γ

(j)i , j = 1, . . . , n− 3, i = 1, . . . , j, as

shown in Figure 6.Let us start with the triangle (0,0)(1,0)(0,1). By (12) we require 1 =

uσ

(1)1uρ(1)1uγ(1)1λ(0,0). Choose u

σ(1)1

= uγ(1)1

= 1 and uρ(1)1

= λ(0,0).

Next consider the triangle(1,0)(0,1)(1,1). We have 1 = uσ

(2)2uγ(2)1λ(0,0)λ

′(0,0),

so choose uσ

(2)2

= 1 and uγ(2)1

= λ(0,0)λ′(0,0). Similarly, setting u

σ(2)1

= uγ(2)2

= 1,



Figure 6. Labels for the vertices and edges of A(n)

uρ(2)1

= λ′(0,0)λ(0,0)λ(1,0) and uρ(2)2

= λ(0,1) then (12) is satisfied for the triangles

(1,0)(2,0)(1,1) and (0,1)(1,1)(0,2).

Continuing in this way we set uγ(k)k

= 1, uγ(k)i

= uρ(k−1)i

λ′(k−i−1,i−1), for

i = 1, . . . , k − 1, and uσ

(k)i

= 1, uρ(k)i

= uρ(k−1)i

λ′(k−i−1,i−1)λ(k−i,i−1), for i =

1, . . . , k, for each k ≤ n−3. Hence, any choice of λ and λ′ will give an equivalentsolution to (13), (14).

For A(∞), we have Perron-Frobenius eigenvectors φ = (φλ1,λ2) given by

φ(λ1,λ2) =[λ1 + 1]q[λ2 + 1]q[λ1 + λ2 + 2]q

[2]q.

Then the rest of the proof follows as for finite n.

Using these cells W we obtain the following representation of the Heckealgebra for A(n). We have written the label for the rows (and columns) infront of each matrix.

U ((λ1,λ2),(λ1,λ2+1)) =(λ1+1,λ2)

(λ1−1,λ2+1)

[λ1+2][λ1+1]

√[λ1][λ1+2]

[λ1+1]√[λ1][λ1+2]

[λ1+1][λ1]

[λ1+1]

,(18)

U ((λ1,λ2),(λ1−1,λ2)) =(λ1−1,λ2+1)

(λ1,λ2−1)

[λ2+2][λ2+1]

√[λ2][λ2+2]

[λ2+1]√[λ2][λ2+2]

[λ2+1][λ2]

[λ2+1]

,(19)

U ((λ1,λ2),(λ1+1,λ2−1))(20)

=(λ1+1,λ2)

(λ1,λ2−1)

[λ1+λ2+3][λ1+λ2+2]

√[λ1+λ2+1][λ1+λ2+3]

[λ1+λ2+2]√[λ1+λ2+1][λ1+λ2+3]

[λ1+λ2+2][λ1+λ2+1][λ1+λ2+2]

.



Let e1, e2, e3 be vectors in the direction of the edges from vertex (λ1, λ2) tothe vertices (λ1 +1, λ2), (λ1−1, λ2 +1), (λ1, λ2−1) respectively, and define aninner-product by ej · ek = δj,k − 1/N . Wenzl [42] constructed representationsof the Hecke algebra, which are given in [14] as:

(21)

λ - λ+ ek

λ+ ej?

- λ+ ej + ek?

= (1− δjl)√sjl(λ′ + ej)sjl(λ′ + ek)

sjl(λ′),

where λ = (λ1, λ2) is a vertex on A(n), λ′ = (λ1 + 1, λ2 + 1), and sjl(λ) =sin((π/n)(ej − el) · λ). Note that this weight is 0 when j = l.

Lemma 5.2. The weights in the representation of the Hecke algebra givenabove for A(n) are identical to those in (21).

Proof. For j = l the result is immediate since there is no triangle λ - λ+ej - λ + 2ej - λ on A(n), and hence the weight in our representationof the Hecke algebra will be zero also. For an arbitrary vertex λ = (λ1, λ2)of A(n), sjl(λ

′) = sin((π/n)(ej − el) · ((λ1 + 1)e1 − (λ2 + 1)e3)). We willshow the result for j = 1, l = 2 (the other cases follow similarly). We haves12(λ

′) = sin((λ1 + 1)π/n) and s12(λ′ + ej) = s12(λ

′ + e1) = sin((λ1 + 2)π/n).We also have s12(λ

′ + e2) = sin(λ1π/n). Then for k = 1, (21) becomes√

sin2((λ1 + 2)π/n)

sin((λ1 + 1)π/n)=

[λ1 + 2]

[λ1 + 1],

whilst for k = 2, (21) becomes√

sin((λ1 + 2)π/n) sin(λ1π/n)

sin((λ1 + 1)π/n)=

√[λ1][λ1 + 2]

[λ1 + 1],

as required.

6. D graphs

The Perron-Frobenius weights for the vertices of A(n) are invariant underthe Z3 symmetry given by rotation by 2π/3. The graph D(n) is obtained fromthe graph A(n) by taking its Z3 orbifold, as illustrated in Figure 7 for n = 9[17]. The Perron-Frobenius weights for the vertices of D(n) are equal to thecorresponding weights in A(n), except that for n = 3k + 3, for integer k ≥ 1,the vertices (k, k)1, (k, k)2 and (k, k)3 (see Figure 8) which come from the fixedpoint (k, k) of A(3k+3) under the rotation whose Perron-Frobenius weights area third of the weight for the vertex (k, k) of A(3k+3). The absolute values |WA|of the cells for A(n) are also invariant under the rotation.

Let n ≥ 5, n 6≡ 0 mod 3. We will find one solution (up to a choice ofphase) for the cells of D(n) by identifying the absolute values |W (A)| for thecells in A(n) with the absolute values |W (D)| for the corresponding cells in D(n)

when taking the orbifold. Each type I frame in D(n) has a corresponding type Iframe in A(n), and similarly for the type II frames. Since the Perron-Frobenius



Figure 7. A(9) and its Z3 orbifold D(9)

weights are the same for A(n) and D(n), these |WD| will certainly satisfy (2)and (3) since the |WA| do. As in the case of A(n), there are no restrictions onthe choice of phase. Then we have the following theorem:

Theorem 6.1. Every orbifold solution for the cells of D(n), n 6≡ 0 mod 3, isequivalent to the solution for which the cells in D(n) are equal to the corre-sponding cells in A(n) given in (13), (14).

Proof. The unitaries ui,j ∈ T, for i, j vertices onD(n), may be chosen systemat-ically as in the proof of Theorem 5.1, beginning with u(k,k),(k,k) =

λ(k,k),(k,k),(k,k)1/3

if n = 3k + 4 or u(k+1,k),(k+1,k) = λ(k+1,k),(k+1,k),(k+1,k)1/3

if n = 3k + 5, and proceeding triangle by triangle.

Now let n = 3k + 3 for some integer k ≥ 1. For q = eiπ/(3k+3), we have[(3k+ 3)/2 + i]q = [(3k+ 3)/2− i]q where i ∈ Z for k even and i ∈ Z + 1

2 for kodd. In particular we will use [2k+ 2 + j] = [k+ 1− j] for j ∈ Z. The Perron-Frobenius weights φ(k,k)i = φ(k,k)/3 = [k + 1]2[2k+ 2]/(3[2]) = [k + 1]3/(3[2]),

i = 1, 2, 3. We again find an orbifold solution for the cells for D(3k+3), exceptfor those which involve the vertices (k, k)i, i = 1, 2, 3, which correspond tothe fixed point (k, k) on the graph A(3k+3). Let γ, γ′ be the two edges inthe double edge of D(3k+3), where γ is the edge from (k, k − 1) to (k − 1, k)and γ′ is the edge from (k, k − 1) to (k + 1, k − 1) in A(3k+3) (see Figure 7).

We will use the notation W(ξ)v,(k,k−1),(k−1,k) to denote the cell for the triangle

v,(k,k−1),(k−1,k) where the edge ξ ∈ γ, γ′ is used, for v = (k − 1, k − 1),(k + 1, k − 2) or (k, k)i, i = 1, 2, 3. Then in particular we have the following:

|W (γ)(k−1,k−1),(k,k−1),(k−1,k) |2 =

[k]2[k + 1]2[2k][2k + 1]

[2]2



=[k]2[k + 1]2[k + 2][k + 3]

[2]2,

|W (γ′)(k+1,k−2),(k,k−1),(k−1,k) |2 =

[k − 1][k][k + 1][k + 2][2k + 1][2k + 2]

[2]2

=[k − 1][k][k + 1]2[k + 2]2

[2]2.

Since γ′ is not an edge used to form the triangle (k−1,k−1),(k,k−1),(k−1,k)

in A(3k+3), we obtain W(γ′)(k−1,k−1),(k,k−1),(k−1,k) = 0. Similarly we obtain

W(γ)(k+1,k−2),(k,k−1),(k−1,k) = 0. The cells involving the vertices (k, k)i coming

from the triplicated vertex (k, k) in A(3k+3) will then be a third of the cor-

responding cells for A(3k+3), since the type I frames(k−1,k)• -

(k,k)i• give

|W (γ)(k−1,k),(k,k)i ,(k,k−1)|2 + |W (γ′)

(k−1,k),(k,k)i ,(k,k−1)|2 = [k][k+ 1]4[k+ 2]/(3[2]) for

i = 1, 2, 3. So

|W (γ)(k−1,k),(k,k)i,(k,k−1)|2 =

1

3|W(k−1,k),(k,k),(k,k−1) |2 =

1

3

[k]2[k + 1]3[k + 2]

[2]2,

|W (γ′)(k−1,k),(k,k)i,(k,k−1)|2 =

1

3|W(k+1,k−1),(k,k),(k,k−1) |2 =

1

3

[k][k + 1]3[k + 2]2

[2]2.

Figure 8. Labels for the graph D(3k+3)

The phase λ of the cell W is the number λ ∈ T such that W = λ|W |.Let λi, λ

′i ∈ T, be the choice of phase for the cells W

(γ)(k−1,k),(k,k)i,(k,k−1),

W(γ′)(k−1,k),(k,k)i ,(k,k−1) respectively. Similarly, let λ

(ξ)(k−1,k−1),(k,k−1),(k−1,k) be

the phase for W(ξ)(k−1,k−1),(k,k−1),(k−1,k) , where ξ ∈ γ, γ′, and Wv1,v2,v3 =

λv1,v2,v3 |Wv1,v2,v3 | for all other triangles v1,v2,v3 of D(3k+3). The type II

frame(k,k−1)• -

-

(k−1,k)• gives the following restriction on the phases λi, λ′i:

(22) λ1λ′1 + λ2λ′2 + λ3λ′3 = 0.

From the type II frame(k,k)i• -

(k,k−1)• (k,k)j• we obtain

Re(λiλ′jλ′iλj) = −1/2 for i 6= j, giving λiλ′i = (−1/2 + εij

√3i/2)λjλ′j , εij ∈



±1. Note that εji = −εij , and substituting for λiλ′i with j = i+ 1 into (22)we find ε12 = ε23 = ε31. Then we have

(23) λiλ′i = (−1

2+ ε

√3i

2)λi+1λ′i+1,

for ε ∈ ±1, i = 1, 2, 3 (mod 3). Then there are two solutions for the cells ofD(3k+3), W and its complex conjugate W . The solution W is the solution tothe graph where we switch vertices (k, k)2 - (k, k)3.

Theorem 6.2. Every orbifold solution for the cells of D(3k+3) is given, upto equivalence, by the inequivalent solutions W or its complex conjugate W ,where W is given by

W(γ)(k−1,k),(k,k)i,(k,k−1) = ǫi

[k]√

[k + 1]3[k + 2]√3 [2]

,

W(γ′)(k−1,k),(k,k)i,(k,k−1) = ǫi

[k + 2]√

[k][k + 1]3√3 [2]

,

W(γ)(k−1,k−1),(k,k−1),(k−1,k) =

[k][k + 1]√

[k + 2][k + 3]]

[2],

W(γ′)(k+1,k−2),(k,k−1),(k−1,k) =

[k + 1][k + 2]√

[k − 1][k]

[2],

W(γ′)(k−1,k−1),(k,k−1),(k−1,k) = W

(γ)(k+1,k−2),(k,k−1),(k−1,k) = 0,

where ǫ1 = 1, ǫ2 = e2πi/3 = ǫ3, and all other cells are equal to the correspondingcells in A(3k+3) given in (13), (14).

Proof. Let W ♯ be any orbifold solution for the cells of D(3k+3). Then W ♯ isgiven, for i = 1, 2, 3, by

W♯(γ)(k−1,k),(k,k)i,(k,k−1) = λ♯i |W

(γ)(k−1,k),(k,k)i ,(k,k−1)|,

W♯(γ′)(k−1,k),(k,k)i,(k,k−1) = λ♯i

′|W (γ′)(k−1,k),(k,k)i ,(k,k−1)|,

W♯(ξ)(k−1,k−1),(k,k−1),(k−1,k) =

λ♯(ξ)(k−1,k−1),(k,k−1),(k−1,k) |W

♯(ξ)(k−1,k−1),(k,k−1),(k−1,k) |,

where ξ ∈ γ, γ′, and W ♯v1,v2,v3 = λ♯v1,v2,v3 |Wv1,v2,v3 | for all other triangles

v1,v2,v3 of D(3k+3), and where the choice of λ♯i , λ♯i′ satisfy condition (23)

with ε = 1. We need to find a family of unitaries uρ for edges ρ 6= γ′ of

D(3k+3), where uγ = (uγ(ξ, ξ′)), ξ, ξ′ ∈ γ, γ′, is a 2 × 2 unitary matrix, and

uρ ∈ T for all other ρ. These unitaries must satisfy (11) and (12), i.e. ǫl =

uµluµ′l(uγ(γ, γ)λ

♯l + uγ(γ, γ

′)λ♯l′) and ǫl = uµluµ′

l(uγ(γ

′, γ)λ♯l + uγ(γ′, γ′)λ♯l

′),for l = 1, 2, 3, and

1 = uσ1uσ2

∑

ξ′

u(ξ, ξ′)λ♯(ξ′)(k−1,k−1),(k,k−1),(k−1,k) ,



1 = uσ′1uσ′

2

∑

ξ′

u(ξ, ξ′)λ♯(ξ′)(k+1,k−2),(k,k−1),(k−1,k) .

For all other triangles (ρ1,ρ2,ρ3)p1,p2,p3 of D(3k+3) we require 1 = uρ1uρ2uρ3λ

♯p1,p2,p3 .

For uγ we choose uγ(γ, γ) = 1, uγ(γ, γ′) = uγ(γ

′, γ) = 0 and uγ(γ′γ′) =

λ♯1λ♯1. We set uµ′

l= 1 and uµl = ǫlλ

♯l , for l = 1, 2, 3, and uσ1 = uσ′

1= 1,

uσ2 = λ♯(γ)(k−1,k−1),(k,k−1),(k−1,k) and uσ′

2= λ

♯(γ′)(k+1,k−2),(k,k−1),(k−1,k) .

Figure 9. Triangles (ρ(1),ρ(2),ρ(3))(i,j),(i−1,j+1),(i,j+1) and (ρ(1)′,ρ(2)′,ρ(3)′)

(i−1,j),(i,j),(i−1,j+1)

For the remaining triangles we proceed as follows. Let m = 2k − 2. For

each triangle (ρ(1),ρ(2),ρ(3))(i,j),(i−1,j+1),(i,j+1) as in Figure 9 (and similarly for triangles

(i,j),(i−1,j+1),(i,j+1)) such that i+j = m, if either uρ(1) or uρ(2) hasn’t yet been

assigned a value we set it to be 1, and set uρ(3) = uρ(1)uρ(2)λ♯(i,j),(i−1,j+1),(i,j+1) .

Next, for each triangle (ρ(1)′,ρ(2)′,ρ(3)′)(i−1,j),(i,j),(i−1,j+1) as in Figure 9 (and similarly

for triangles (i+1,j−1),(i,j),(i+1,j)) such that i + j = m, if either uρ(1)′ oruρ(2)′ hasn’t yet been assigned a value we set it to be 1, and set uρ(3)′ =

uρ(1)′uρ(2)′λ♯(i−1,j),(i,j),(i−1,j+1) . We then set m = 2k − 3 and repeat the above

steps. Continuing in this way, for m = 2k− 4, . . . , 3, we find the required uni-taries uρ. The proof for the uniqueness of the complex conjugate solutioncan be shown similarly.

For the solutions W and W to be equivalent, we require unitaries as abovesuch that

ǫl = uµluµ′l(uγ(γ, γ)ǫl +

√[k + 2]√

[k]uγ(γ, γ

′)ǫl),

ǫl = uµluµ′l(

√[k]√

[k + 2]uγ(γ

′, γ)ǫl + uγ(γ′, γ′)ǫl),

for l = 1, 2, 3. This forces uγ(γ, γ) = uγ(γ′, γ′) = 0, uγ(γ, γ

′) =√

[k]/√

[k + 2]

and uγ(γ′, γ) =

√[k + 2]/

√[k]. But then uγ is not a unitary.

Using the cells W we obtain the following representation of the Hecke alge-bra for D(3k+3), we use the notation v(γ) if the path uses the edge γ, where v



is a vertex of D(3k+3):

U ((k−1,k−1),(k−1,k)) =

(k,k−1)(γ)

(k,k−1)(γ′)

(k−2,k)

[k+1][k] 0

√[k−1][k+1]

[k]

0 0 0√[k−1][k+1]

[k] 0 [k−1][k]

= U ((k,k−1),(k−1,k−1)) with rows labelled by (k − 1, k)(γ), (k − 1, k)(γ′),

U ((k+1,k−2),(k−1,k)) =

(k,k−1)(γ)

(k,k−1)(γ′)

(k−2,k)

0 0 0

0 [k+1][k+2]

√[k+1][k+3]

[k+2]

0

√[k+1][k+3]

[k+2][k+3][k+2]

= U ((k,k−1),(k+1,k−2)) with rows labelled by (k − 1, k)(γ), (k − 1, k)(γ′),

(k, k − 2),

U ((k,k−1),(k,k)i) =(k−1,k)(γ)

(k−1,k)(γ′)

[k][k+1] ǫi

√[k][k+2]

[k+1]

ǫi

√[k][k+2]

[k+1][k+2][k+1]

,

= U ((k,k)i,(k−1,k)) with rows labelled by (k, k − 1)(γ), (k, k − 1)(γ′),

U ((k−1,k),(k,k−1)) =

(k,k)1

(k,k)2

(k,k)3

(k−1,k−1)

(k+1,k−2)

[2][k + 1]a ǫa ǫa b c

ǫa [2][k + 1]a ǫa ǫ2b ǫ2c

ǫa ǫa [2][k + 1]a ǫ2b ǫ2c

b ǫ2b ǫ2b[k+3][k+2] 0

c ǫ2c ǫ2c 0 [k−1][k]

,

where ǫ = ǫ2[k] + ǫ2[k + 2] and

a =[k + 1]

3[k][k + 2], b =

√[k + 1][k + 3]√

3 [k + 2], c =

√[k − 1][k + 1]√

3 [k].

Another representation of the Hecke algebra is given by taking the complexconjugates of the weights in the representation above.

In [25], Fendley gives Boltzmann weights for D(6), which at criticality andwith the parameter u = 1, give a representation of the Hecke algebra. Howeverthese Boltzmann weights are not equivalent to the representation of the Heckealgebra using the cells W or W . To see this, we use a similar labelling for thegraph D(6) as in [25]- see Figure 10.

Consider the weight [U(3r,2)

]γ,γ′ , where we label the rows and columns byγ, γ′ to denote which edge from 1 to 2 is used for the path of length 2 from3r to 2, r = 0, 1, 2, and the weight U is the complex conjugate of that givenabove, i.e. it is the weight given by the solution W for the cells of D(6). Then



Figure 10. Labelling the graph D(6)

for equivalence we require a unitary u3r,1 ∈ T and a 2 × 2 unitary matrix uγsuch that

ǫr2

√[3]

[2]= |u3r,1|2

(uγ(γ, γ)uγ(γ′, γ)

1

[2]+ uγ(γ, γ)uγ(γ′, γ′)ǫ

r2

√[3]

[2]

+uγ(γ, γ′)uγ(γ′, γ)ǫ

r2

√[3]

[2]+ uγ(γ, γ

′)uγ(γ′, γ′)[3]

[2]

).(24)

Since uγ is independent of r, for (24) to be satisfied for each r = 0, 1, 2, we

require uγ(γ, γ)uγ(γ′, γ′) = 1 and the other terms to be zero, which givesuγ(γ, γ

′) = uγ(γ′, γ) = 0 and uγ(γ

′, γ′) = (uγ(γ, γ))−1. But now if we consider

the weight [U(1,3r)

]γ,γ′ , with u2,3r ∈ T, we have

ǫr2

√[3]

[2]= |u2,3r |2

(uγ(γ, γ)uγ(γ′, γ)

1

[2]+ uγ(γ, γ)uγ(γ′, γ′)ǫ

r2

√[3]

[2]

+uγ(γ, γ′)uγ(γ′, γ)ǫ

r2

√[3]

[2]+ uγ(γ, γ

′)uγ(γ′, γ′)[3]

[2]

),

but [U(1,3r)

]γ,γ′ = ǫr2

√[3]

[2] , for r = 0, 1, 2. We obtain a similar contradiction

when considering the weights U defined using the solution W for the cells.

Suppose however that the Boltzmann weight denoted by W(e1,f3r)e2,e2 in [25] is

the complex conjugate of that given. Then the Boltzmann weights at criticalityof Fendley [25] are equivalent to the representation of the Hecke algebra givenby the solution W for the cells of D(6). We choose a family of unitaries u0,1 =u2,0 = u2,3r = 1, u3r,1 = ǫr2, r = 0, 1, 2, and choose uγ to be the 2× 2 identitymatrix.

7. A∗ graphs

The infinite series of graphs A(n)∗ are illustrated in Figure 11. The graphsA(2n+1)∗ and A(2n)∗ are slightly different.

First we consider the graphs A(2n+1)∗. The Perron-Frobenius weights onthe vertices are given by φi = [2i− 1], i = 1, . . . , n.



Figure 11. A(n)∗ for n = 4, 5, 6, 7, 8, 9

Theorem 7.1. There is up to equivalence a unique set of cells for A(2n+1)∗,n <∞, given by

Wi−1,i,i =

√[i][2i− 3][2i− 1]√

[i− 1], i = 2, . . . , n,

Wi,i,i+1 =

√[i− 1][2i− 1][2i+ 1]√

[i], i = 2, . . . , n− 1,

Wi,i,i = (−1)i+1 [2i− 1]√[i− 1][i]

, i = 2, . . . , n.

Proof. Using (2), (3) we obtain

|Wi−1,i,i|2 =[i][2i− 3][2i− 1]

[i− 1], i = 2, . . . , n,(25)

|Wi,i,i+1|2 =[i− 1][2i− 1][2i+ 1]

[i], i = 2, . . . , n− 1,(26)

|Wi,i,i|2 =[2i− 1]2

[i− 1][i], i = 2, . . . , n.(27)

Let Wi,j,k = λi,j,k|Wi,j,k| for λi,j,k ∈ T. From type II frames we have therestriction

(28) λ3i,i,i+1λi+1,i+1,i+1 = −λ3

i,i+1,i+1λi,i,i,

for i = 2, . . . , n − 1. Let W ♯i,j,k = λ♯i,j,k|Wi,j,k| be any other solution to

the cells, where the λ♯ satisfy (28). We need to find a family of unitariesui,j, where ui,j is the unitary for the edge from vertex i to vertex j on

A(2n+1)∗, which satisfy (12), i.e. −1 = u32l,2lλ

♯2l,2l,2l for l = 1, . . . , ⌊n/2⌋,

and 1 = uiujukλ♯i,j,k for all other triangles i,j,k. We choose u1,2 = 1,

u2,1 = −(λ♯2,2,2)1/3λ♯1,2,2, u2,2 = −(λ♯2,2,2)

1/3, and for i = 2, . . . , n−1, ui,i+1 = 1

ui+1,i = −(λ♯2,2,2)1/3λ♯2,3,3λ

♯3,4,4 · · ·λ♯i−1,i,iλ

♯2,2,3λ

♯3,3,4 · · ·λ♯i,i,i+1, and ui+1,i+1 =

−(λ♯2,2,2)1/3λ♯2,2,3λ

♯3,3,4 · · ·λ♯i,i,i+1λ

♯2,3,3λ

♯3,4,4 · · ·λ♯i,i+1,i+1.

For A(2n+1)∗, the above cells W give the following representation of theHecke algebra:

U (i,i+1) =i

i+1

[i−1][i]

√[i−1][i+1]

[i]√[i−1][i+1]

[i][i+1][i]

,



U (i,i−1) =i−1

i

[i−2][i−1]

√[i−2][i]

[i−1]√[i−2][i]

[i−1][i]

[i−1]

,

U (i,i) =

i−1

i

i+1

[i][2i−3][i−1][2i−1]

(−1)i+1√

[2i−3]

[i−1]√

[2i−1]

√[2i−3][2i+1]

[2i−1]

(−1)i+1√

[2i−3]

[i−1]√

[2i−1]

1[i−1][i]

(−1)i+1√

[2i+1]

[i]√

[2i−1]√[2i−3][2i+1]

[2i−1]

(−1)i+1√

[2i+1]

[i]√

[2i−1]

[i−1][2i+1][i][2i−1]

.

In [4], Behrend and Evans give Boltzmann weights

W

(a d

b c

∣∣∣∣∣u),

which at criticality, with u = 1, give a representation of the Hecke algebra.(Note, these Boltzmann weights are not to be confused with the Ocneanu cellsW .)

Lemma 7.2. The weights in the representation of the Hecke algebra givenabove for A(2n+1)∗ are equivalent to the Boltzmann weights at criticality givenby Behrend-Evans in [4].

Proof. To make our notation the same as that of [4] one replaces i with (a +1)/2. Then it is easily checked that the absolute values of our weights givenabove are equal to those for the Boltzmann weights in [4], setting q = 0, in allbut a few cases. We will show that the absolute values in these other cases arealso equal. For [U (i,i)]i+1,i+1, the Boltzmann weight in [4] is

[a+ 2]− [a+ 2]/[a]

[a+ 1]=

[a+ 2]

[a][a+ 1]([a]− [1]) =

[a+ 2]

[a][a+ 1]

[ 12 (a− 1)][a+ 1]

[ 12 (a+ 1)],

which is equal to our weight, and similarly for [U (i,i)]i−1,i−1. For [U (i,i)]i,i wehave to do the most work. From [4] its value is

(29)1

[3]

([2]− [a+ 2][12 (a− 5)]

[a][12 (a+ 1)]− [a− 2][12 (a+ 5)]

[a][12 (a− 1)]

).

Writing this expression over a common denominator, and using (1), we canwrite the numerator as

[2][a]([2] + [4] + · · ·+ [a− 1])− [a+ 2]([3] + [5] + · · ·+ [a− 4])

− [a− 2]([3] + [5] + · · ·+ [a+ 2])

= [a]([1] + [3] + [3] + [5] + · · ·+ [a− 2] + [a])

− ([a+ 2] + [a− 2])([3] + [5] + · · ·+ [a− 4])

− [a− 2]([a− 2] + [a] + [a+ 2])

= [a] + (2[a]− [a+ 2]− [a− 2])([3] + [5] + · · ·+ [a− 4] + [a− 2])

+ [a]2 − [a− 2][a]



= [a] + ([a]− [a+ 2])([3] + [5] + · · ·+ [a− 2])

+ ([a]− [a− 2])([3] + [5] + · · ·+ [a− 2] + [a])

= [a] + [(a− 3)/2][(a+ 1)/2]([a]− [a+ 2])

+ [(a− 1)/2][(a+ 3)/2]([a]− [a− 2]).

Now

[(a− 3)/2][(a+ 1)/2]([a]− [a+ 2])

= [(a− 3)/2]([(a+ 1)/2] + [(a+ 5)/2] + · · ·+ [(3a− 1)/2]

− [(a+ 5)/2]− [(a+ 9)/2]− · · · − [(3a+ 3)/2])

= [(a− 3)/2]([(a+ 1)/2]− [(3a+ 3)/2])

= [3] + [5] + · · ·+ [a− 2]− [a+ 4]− [a+ 6]− · · · − [2a− 1],

and

[(a− 1)/2][(a+ 3)/2]([a]− [a− 2])

= [(a− 1)/2]([(a+ 1)/2] + [(a+ 3)/2] + · · ·+ [(3a+ 1)/2]

− [(a− 5)/2]− [(a− 1)/2]− · · · − [(3a− 3)/2])

= [(a− 1)/2]([(3a+ 1)/2]− [(a− 5)/2])

= [a+ 2] + [a+ 4] + · · ·+ [2a− 1]− [3]− [5]− · · · − [a− 4].

Then we find that the numerator is given by [a] + [a− 2]+ [a+2] = [3][a], and(29) becomes

[3][a]

[3][a][12 (a− 1)][12 (a+ 1)]=

1

[ 12 (a− 1)][12 (a+ 1)]

as required. To show equivalence, we need unitaries ui,j ∈ T, for vertices i, j

of A(n)∗ such that

1 = ui,iui+1,i+1, 1 = ui,iui−1,i−1, −1 = ui,i−1ui−1,iui,i+1ui+1,i,

(−1)i = u2i,iui,i+1ui+1,i, (−1)i+1 = u2

i,iui,i−1ui−1,i.

Then we set ui,i = 1 for all i, and for m = 0, . . . , (n − 2)/2, u2m+1,2m =u2m,2m+1 = u2m+2,2m+1 = 1 and u2m+1,2m+2 = −1.

For the graphsA(4n)∗ (illustrated in Figure 11) the Perron-Frobenius weightson the vertices are given by φi = [2i]/[2], i = 1, . . . , 2n − 1. There are nowtwo solutions W+, W− for the cells for A(4n)∗, which are not equivalent since|W+| 6= |W−| and the graph A(4n)∗ does not contain any multiple edges.

Theorem 7.3. The cells for A(4n)∗, n < ∞, are given, up to equivalence, bythe inequivalent solutions W+, W−:

W±i,i,i+1 =

√[2i][2i+ 2]

[2]√

[2i+ 1]

√[2i]∓ [1], i = 1, . . . , 2n− 2,



W±i,i+1,i+1 =

√[2i][2i+ 2]

[2]√

[2i+ 1]

√[2i+ 2]± [1], i = 1, . . . , 2n− 2,

W±i,i,i =

(−1)i+1

√[2i]

[2]√

[2i− 1][2i+ 1]

√[2][2i]± [4i], i = 1, . . . , n− 1,

(−1)n+1 [2n]√[2][2n− 1][2n+ 1]

, i = n,

(−1)i+1

√[2i]

[2]√

[2i− 1][2i+ 1]

√[2][2i]∓ [8n− 4i],

i = n+ 1, . . . , 2n− 1.

Proof. The proof follows in a similar way to the A(2n+1)∗ case.

For the graphs A(4n+2)∗ (illustrated in Figure 11) the Perron-Frobeniusweights on the vertices are again given by φi = [2i]/[2], i = 1, . . . , 2n. Thereare again two inequivalent solutions W+, W− for the cells of A(4n+2)∗.

Theorem 7.4. The cells for A(4n+2)∗, n < ∞, are given, up to equivalence,by the inequivalent solutions W+, W−:

W±i,i,i+1 =

√[2i][2i+ 2]

[2]√

[2i+ 1]

√[2i]∓ [1], i = 1, . . . , 2n− 1,

W±i,i+1,i+1 =

√[2i][2i+ 2]

[2]√

[2i+ 1]

√[2i+ 2]± [1], i = 1, . . . , 2n− 1,

W±i,i,i =

(−1)i+1

√[2i]

[2]√

[2i− 1][2i+ 1]

√[2][2i]± [4i], i = 1, . . . , n,

(−1)i+1

√[2i]

[2]√

[2i− 1][2i+ 1]

√[2][2i]∓ [8n+ 4− 4i],

i = n+ 1, . . . , 2n.

Proof. The proof again follows in a similar way to the A(2n+1)∗ case.

ForA(2n)∗, the cellsW+ above give the following representation of the Heckealgebra:

U (i,i+1) =i

i+1

[2i]−[1][2i+1]

√([2i]−[1])([2i+2]+[1])

[2i+1]√([2i]−[1])([2i+2]+[1])

[2i+1][2i+2]+[1]

[2i+1]

,

U (i,i−1) =i−1

i

[2i−2]−[1][2i−1]

√([2i−2]−[1])([2i]+[1])

[2i−1]√([2i−2]−[1])([2i]+[1])

[2i−1][2i]+[1][2i−1]

,



U (i,i) =

i−1

i

i+1

[2i−2]([2i]+[1])[2i][2i+1] (−1)i+1√xa+

√[2i−2][2i−1][2i+2]

[2i]√

[2i+1]

(−1)i+1√xa+ x (−1)i+1√xa−√[2i−2][2i−1][2i+2]

[2i]√

[2i+1](−1)i+1√xa− [2i+2]([2i]−[1])

[2i][2i+1]

,

where, a± = [2i∓ 2]([2i]± [1])/[2i][2i+ 1], and for m > 0, if n = 2m,

x =

[2][2i]+[4i][2i−1][2i][2i+1] for i = 1, . . . ,m− 1,

[2][2m−1]2 for i = m,

[2][2i]−[4n−4i][2i−1][2i][2i+1] for i = m+ 1, . . . , 2m− 1,

,

and if n = 2m+ 1,

x =

[2][2i]+[4i][2i−1][2i][2i+1] for i = 1, . . . ,m,

[2][2i]−[4n−4i][2i−1][2i][2i+1] for i = m+ 1, . . . , 2m,

.

Lemma 7.5. The weights in the representation of the Hecke algebra givenabove for A(2n)∗ are equivalent to the Boltzmann weights at criticality given byBehrend-Evans in [4].

Proof. To make our notation the same as that of [4] one replaces i with a/2. Tosee that the absolute values of our weights are equal to those of the Boltzmannweights in [4] one needs the following relations on the quantum numbers:

[2i] + [1] =[2i+ 1]q′ [4i+ 2]q′

[2i− 1]q′, [2i]− [1] =

[2i− 1]q′ [4i+ 2]q′

[2i+ 1]q′,

where q′ =√q (q = eiπ/n). Again, a bit more work is required for [U (i,i)]i,i.

For equivalence we make the same choice of (ui,j)i,j as for A(2n+1)∗.

8. D∗ graphs

The graphs D(n)∗ are illustrated in Figure 12. We label its vertices by il, jland kl, l = 1, . . . , ⌊(n−1)/2⌋, which we have illustrated in Figure 12 for n = 9.

We consider first the graphs D(2n+1)∗. The Perron-Frobenius weights areφil = φjl = φkl = [2l − 1], l = 1, . . . , n. Since the graph has a Z3 symmetry,we will seek Z3-symmetric solutions (up to choice of phase), i.e. |Wip,jq,kr |2 =

|Wiq ,jr ,kp |2 = |Wir ,jp,kq |2 =: |Wp,q,r|2, p, q, r ∈ 1, . . . , n. Using this notation,we have the following equations from type I frames:

|W1,2,2|2 = [2][3],(30)

|Wl,l,l+1|2 + |Wl,l+1,l+1|2 = [2][2l− 1][2l+ 1], l = 2, . . . , n− 1,(31)

|Wl−1,l,l|2 + |Wl,l,l|2 + |Wl,l,l+1|2 = [2][2l− 1]2, l = 2, . . . , n− 1,(32)

|Wn−1,n,n|2 + |Wn,n,n|2 = [2]3,(33)



Figure 12. D(n)∗ for n = 6, 7, 8, 9

and from type II frames we have:

(34) |Wl−1,l,l|2|Wl,l,l+1|2 = [2l− 3][2l− 1]2[2l + 1],

for l = 2, . . . , n− 1, and

(35) |Wl−1,l,l|2(1

[2l − 3]|Wl−1,l−1,l|2 +

1

[2l − 1]|Wl,l,l|2) = [2l− 3][2l− 1]2,

for l = 2, . . . , n, which are exactly those for the type I and type II frames forthe graph A(2n+1)∗. Since the Perron-Frobenius weights and Coxeter numberare also the same as for A(2n+1)∗, the cells |Wp,q,r,| follow.

From the type II frame consisting of the vertices il, jl, il+1 and jl+1 we havethe following restriction on the choice of phase

λil,jl,kl+1λil,jl+1,klλil+1,jl,klλil+1,jl+1,kl+1

(36)

= −λil,jl,klλil,jl+1,kl+1λil+1,jl,kl+1

λil+1,jl+1,kl .

Theorem 8.1. Every Z3-symmetric solution for the cells W of D(2n+1)∗, n <∞, is equivalent to the solution

Wi1,j2,k2 = Wi2,j1,k2 = Wi2,j2,k1 =√

[2][3],

Wil,jl+1,kl+1= Wil+1,jl,kl+1

= Wil+1,jl+1,kl =

√[l + 1][2l− 1][2l+ 1]√

[l],

Wil,jl,kl+1= Wil,jl+1,kl = Wil+1,jl,kl =

√[l − 1][2l− 1][2l+ 1]√

[l],

Wil,jl,kl = (−1)l+1 [2l − 1]√[l − 1][l]

, Win,jn,kn = (−1)n+1 [2n− 1]√[n− 1][n]

,

for l = 2, . . . , n− 1.

Proof. Let W ♯ be any Z3-symmetric solution for the cells of D(2n+1)∗, wherethe choice of phase satisfies the condition (36). Since D(2n+1)∗ does not contain

any multiple edges, we must have |W ♯ijk | = |Wijk | for every triangle ijk of



D(2n+1)∗. We need to find a family of unitaries up,q, where up,q is the uni-

tary for the edge from vertex p to vertex q on D(2n+1)∗, which satisfy (12), i.e.

−1 = ui2l,j2luj2l,k2luk2l,i2lλ♯i2l,j2l,k2l

for the trianglei2l,j2l,k2l , l = 1, . . . , ⌊n/2⌋,and 1 = up1up2up3λp1,p2,p3 for all other triangles on D(2n+1)∗. For triangles

involving the outermost vertices, we require that 1 = ui1,j2uj2,k2uk2,i1λ♯i1,j2,k1

,

1 = ui2,j1uj1,k2uk2,i2λ♯i2,j1,k2

, 1 = ui2,j2uj2,k1uk1,i2λ♯i2,j2,k1

and also −1 =

ui2,j2uj2,k2uk2,i2λ♯i2,j2,k2

. So we choose ui1,j2 = uj1,k2 = uk1,i2 = uj2,k2 =

uk2,i2 = 1, ui2,j1 = λ♯i2,j1,k2 , uk2,i1 = λ♯i1,j2,k2 , ui2,j2 = −λ♯i2,j2,k2 and uj2,k1 =

−λ♯i2,j2,k2λ♯i2,j2,k1

. Next consider the equations 1 = ui2,j3uj3,k2uk2,i2λ♯i2,j3,k2

,


and 1 = ui2,j2uj2,k3uk3,i2λ♯i2,j2,k3

. We make the

following choices: ui2,j3 = uj2,k3 = uk2,i3 = 1, ui3,j2 = λ♯i3,j2,k2 , uj3,k2 =

λ♯i2,j3,k2 and uk3,i2 = −λ♯i2,j2,k2λ♯i2,j2,k3

. Next we consider the equations


= −uj3,k3λ♯i2,j2,k2λ♯i2,j2,k3

λ♯i2,j3,k3 ,


= uk3,i3λ♯i3,j2,k2

λ♯i3,j2,k3 ,


= ui3,j3λ♯i2,j3,k2

λ♯i3,j3,k2 .

We make the choices ui3,j3 = λ♯i2,j3,k2λ♯i3,j3,k2

, uk3,i3 = λ♯i3,j2,k2λ♯i3,j2,k3

and

uj3,k3 = −λ♯i2,j2,k3λ♯i2,j2,k2

λ♯i2,j3,k3 . Then ui3,j3uj3,k3uk3,i3λ♯i3,j3,k3

=

−λ♯i2,j3,k2λ♯i3,j3,k2

λ♯i2,j2,k3λ♯i2,j2,k2

λ♯i2,j3,k3λ♯i3,j2,k2

λ♯i3,j2,k3 = −1, by (36), as re-quired. Continuing in this way we are done.

For D(2n+1)∗, the Hecke representation for the cells W above is given bythe Hecke representation for A(2n+1)∗, where [U (il,kr)]jm,jp = [U (jl,ir)]km,kp =

[U (kl,jr)]im,ip are given by the weights [U (l,r)]m,p for A(2n+1)∗, for any l,m, p, rallowed by the graph.

We now consider the graphs D(2n)∗. The Perron-Frobenius weights areφil = φjl = φkl = [2l]/[2], and we again assume |Wip,jq,kr |2 = |Wiq ,jr,kp |2 =

|Wir ,jp,kq |2 =: |Wp,q,r|2, where p, q, r ∈ 1, . . . , n − 1. Then as for D(2n+1)∗,

the Z3-symmetric solution for the cells follows from the solution for A(2n)∗,and we have the same restriction (36) on the choice of phase. So we have

Theorem 8.2. For n <∞, the Z3-symmetric solution for the cells of D(4n)∗

are given by

W±il,jl,kl+1= W±il,jl+1,kl

= W±il+1,jl,kl=

√[2l][2l+ 2]

[2]√

[2l+ 1]

√[2l]∓ [1],

l = 2, . . . , 2n− 2,



W±il,jl+1,kl+1= W±il+1,jl,kl+1

= W±il+1,jl+1,kl=

√[2l][2l+ 2]

[2]√

[2l+ 1]

√[2l+ 2]± [1],

l = 1, . . . , 2n− 2,

W±il,jl,kl =

(−1)l+1

√[2l]

[2]√

[2l− 1][2l+ 1]

√[2][2l]± [4l], l = 1, . . . , n− 1,

(−1)n+1 [2n]√[2][2n− 1][2n+ 1]

, l = n,

(−1)l+1

√[2l]

[2]√

[2l− 1][2l+ 1]

√[2][2l]∓ [8n− 4l],

l = n+ 1, . . . , 2n− 1,

and the Z3-symmetric solution for the cells of D(4n+2)∗ are

W±il,jl,kl+1= W±il,jl+1,kl

= W±il+1,jl,kl=

√[2l][2l+ 2]

[2]√

[2l+ 1]

√[2l]∓ [1],

l = 2, . . . , 2n− 1,

W±il,jl+1,kl+1= W±il+1,jl,kl+1

= W±il+1,jl+1,kl=

√[2l][2l+ 2]

[2]√

[2l+ 1]

√[2l+ 2]± [1],

l = 1, . . . , 2n− 1,

W±il,jl,kl =

(−1)l+1

√[2l]

[2]√

[2l− 1][2l+ 1]

√[2][2l]± [4l], l = 1, . . . , n,

(−1)l+1

√[2l]

[2]√

[2l− 1][2l+ 1]

√[2][2l]∓ [8n+ 4− 4l],

l = n+ 1, . . . , 2n.

The uniqueness of these solutions follows in the same way as for D(2n+1)∗.If W+ is a solution for the cells of D(2n)∗, then W− is a solution for the cells ofthe graph where we switch vertices il - in−l, jl - jn−l and kl - kn−l,for all l = 1, . . . , n− 1.

For D(2n)∗, the Hecke representation for the cells W+ above is given bythe Hecke representation for A(2n)∗, where [U (il,kr)]jm,jp = [U (jl,ir)]km,kp =

[U (kl,jr)]im,ip are given by the weights [U (l,r)]m,p for A(2n)∗, for any l,m, p, rallowed by the graph.

In [14], di Francesco and Zuber gave a representation of the Hecke algebrafor the graph D(6)∗, with the absolute values of the weights there equal to thosefor our weights given above. The two Hecke representations are not identicalas the weights in [14] involve the complex variable i. However it has not beenpossible to determine whether or not the two representations are equivalent asthere are known to be a number of typographical errors in the representationin [14].



9. E(8)

We will label the vertices of the exceptional graph E(8) in the following way.We will label the six outmost vertices by il and the six inmost vertices byjl, l = 1, . . . , 6, such that there are edges from il to jl and from jl to il+1.The Perron-Frobenius weights on the vertices are φil = 1, φjl = [3]. With

[a] = [a]q, q = eiπ/8, we have [4]/[2] =√

2.

Figure 13. E(8) and its Z3 orbifold E(8)∗

We will again use the notation Wi,j,k for W (i,j,k). Then from the type Iframes on the graph we have the following equations:

|Wil,jl,jl−1|2 = [2]φilφjl = [2][3],

|Wil,jl,jl−1|2 + |Wjl+1,jl,jl−1

|2 + |Wjl,jl−1,jl−2|2 = [2]φjlφjl−1

= [2][3]2.

Then |Wjl+1,jl,jl−1|2 + |Wjl,jl−1,jl−2

|2 = [3][4]. Since there is a Z6 symmetry of

E(8) we assume |Wjl+1,jl,jl−1|2 = |Wjk+1,jk,jk−1

|2 for all k, l, giving

|Wjl+1,jl,jl−1|2 =

1

2[3][4] =

[2]2[3]

[4].

The Z6 symmetry of the cells can be deduced from equation (37). Finally, for

the type I framesjl• -

jl+2• we have |Wjl+2,jl+1,jl |2 + |Wjl,jl+2,jl+4|2 = [2][3]2

giving

|Wjl,jl+2,jl+4|2 = [2][3]2 − [2]2[3]

[4]=

[2]2[3]2

[4].

Let

Wil,jl,jl−1= λil

√[2][3], l = 1, . . . , 6,

Wjl,jl−1,jl−2= λ

(1)jl

[2]√

[3]√[4]

, l = 1, . . . , 6,

Wjl,jl+2,jl+4= λ

(2)jl

[2][3]√[4], l = 1, 2.



The only type II frames that yield anything new are those for the frameinvolving the vertices jl−2, jl−3(= jl+3), jl+1 and jl:

0 = φ−1jl−1

Wjl−2,jl−1,jlWjl+1,jl,jl−1Wjl−1,jl+1,jl+3

Wjl−1,jl−2,jl−3

+ φ−1jl+2

Wjl−2,jl,jl+2Wjl+2,jl+1,jlWjl+3,jl+2,jl+1

Wjl−2,jl−3,jl+2

=[2]4√

[3]3

[4]2λ

(1)jlλ

(1)jl+2

λ(1)jl+4

λ(2)jl−1

+[2]4√

[3]3

[4]2λ

(1)jl−1

λ(1)jl+1

λ(1)jl+3

λ(2)jl,(37)

which for any l = 1, . . . , 6 gives

(38) λ(1)j1λ

(1)j3λ

(1)j5λ

(2)j2

= −λ(1)j2λ

(1)j4λ

(1)j6λ

(2)j1.

From the type II frame above we see that there must be a Z6 symmetry onthe cells, |Wjl+1,jl,jl−1

|2 = |Wjk+1,jk,jk−1|2 for all k, l, is correct since otherwise

the coefficients of the two terms in equation (37) would be different, and (38)would be

λ(1)j1λ

(1)j3λ

(1)j5λ

(2)j2

= −cλ(1)j2λ

(1)j4λ

(1)j6λ

(2)j1,

for some constant c ∈ R with |c| 6= 1, which is impossible.

Theorem 9.1. There is up to equivalence a unique set of cells for E(8) given by

Wil,jl,jl−1=√

[2][3], Wjl,jl−1,jl−2=

[2]√

[3]√[4]

, l = 1, . . . , 6,

Wj1,j3,j5 =[2][3]√

[4], Wj2,j4,j6 = − [2][3]√

[4].

Proof. Let W ♯ be any solution for the for the cells for E(8), where the choiceof phase satisfies the condition (38). We need to find a family of unitariesup,q, where up,q is the unitary for the edge from vertex p to vertex q on E(8),

which satisfy (12), i.e. −1 = uj2,j4uj4,j6uj6,j2λ(2)j2

for the triangle j2,j4,j6 ,and 1 = up1up2up3λp1,p2,p3 for all other triangles, where λp1,p2,p3 is the phase

associated to triangle p1,p2,p3 . We make the choices uil,jl = ujl,jl−1λil ,

ujl,jl+1= 1 for l = 1, . . . , 6, uj2,j1 = uj5,j4 = 1, uj1,j6 = λ

(1)j2

, uj3,j2 =

λ(1)j2λ

(1)j6λ

(2)j1λ

(1)j1λ

(1)j3

, uj4,j3 = λ(1)j5

, uj6,j5 = λ(1)j6

, uj3,j5 = uj4,j6 = uj6,j2 = 1,

uj1,j3 = λ(1)j2λ

(1)j2λ

(1)j6λ

(2)j1

, uj2,j4 = λ(1)j2λ

(1)j3λ

(1)j5λ

(1)j2λ

(1)j4λ

(1)j6λ

(2)j1

and uj5,j1 =

λ(1)j2λ

(1)j6λ

(1)j1

.

For E(8), the above cells W give the following representation of the Heckealgebra:

U (il,jl−1) = U (jl,il) = [2],

U (jl,jl−2) =jl−1

jl+2

1[2]

(−1)l+1√

[3]

[2]

(−1)l+1√

[3]

[2][3][2]

,



U (jl,jl+1) =

jl−1

jl+2

il+1

1[2]

1[2]

1√[3]

1[2]

1[2]

1√[3]

1√[3]

1√[3]

[2][3]

,

for l = 1, . . . , 6 (mod 6). This representation is identical to that given by diFrancesco-Zuber in [14]. (The representation in [14] is given for the graphE(8)∗, and the representation for E(8) is obtained by an unfolding of the graphE(8)∗.)

10. E(8)∗

We will label the vertices of the graph E(8)∗ as in Figure 13. The Perron-Frobenius weights for E(8)∗ are φ1 = φ4 = 1, φ2 = φ3 = [3]. As with thegraphs A(n) and E(8) we easily find |W123|2 = [2][3] and |W234|2 = [2][3]. Then

by the type II frame1• -

2• 2• we have [3]−1|W123|2|W223|2 = [3]2,

and so |W223|2 = [3]2/[2]. Similarly |W233|2 = [3]2/[2]. From the type I frame2• -

2• we get |W222|2 + |W223|2 = [2][3]2, giving |W222|2 = [3]3/[2], andsimilarly |W333|2 = [3]3/[2]. Let Wijk = λijk|Wijk |. Then from the type IIframe consisting of the vertices 2,2,3,3 we obtain the following restriction onthe choice of phase:

(39) λ222λ3233 = −λ333λ

3223.

Theorem 10.1. There is up to equivalence a unique set of cells for E(8)∗ givenby

W123 = W234 =√

[2][3],

W223 = W233 =[3]√[2],

W222 =

√[3]3√[2]

, W333 = −√

[3]3√[2]

.

Proof. Let W ♯ be any solution for the cells for E(8)∗, where the choice of phasesatisfies the condition (39). We need to find a family of unitaries up,q, where

up,q is the unitary for the edge from vertex p to vertex q on E(8)∗, which satisfy(12), i.e. −1 = u3

3,3λ333 for the triangle 3,3,3, and 1 = ui,juj,kuk,iλijk forall other triangles, where λijk is the phase associated to triangle i,j,k. We

choose u3,1 = u3,2 = u4,3 = 1, u2,4 = λ234, u3,3 = −λ333

13 , u2,3 = −λ

13333λ233,

u1,2 = −λ233λ123λ333

13 and u2,2 = −λ233λ223λ333

13 .

For E(8)∗, the above cells W give the following Hecke representation:

U (1,3) = U (2,1) = U (3,4) = U (4,2) = [2],



U (2,2) =3

2

1[2]

√[3]

[2]√[3]

[2][3][2]

,

U (3,3) =2

3

1[2] −

√[3]

[2]

−√

[3]

[2][3][2]

,

U (2,3) =

2

3

4

1[2]

1[2]

1√[3]

1[2]

1[2]

1√[3]

1√[3]

1√[3]

[2][3]

.

= U (3,2) with rows labelled by 2, 3, 1.

This representation is identical to that given by di Francesco-Zuber in [14].

11. E(12)2

We label the vertices and edges of the graph E(12)2 as in Figure 14. The

Perron-Frobenius weights for E(12)2 are

φi = 1, φj = φk = [3], φpl =[2]3

[4], φql = φrl =

[2][3]

[4], l = 1, 2, 3.

Figure 14. E(12)1 and E(12)

2

Let Wv1,v2,v3 = λv1,v2,v3 |Wv1,v2,v3 | for vertices v1, v2, v3 of E(12)2 . The type

II frames consisting of the vertices pl, k, pl−1 and rl give a restriction on thephases λv1,v2,v3 :



0 = φ−1ql−1

Wpl−1,ql−1,rlWpl−1,ql−1,kWpl,ql−1,kWpl,ql−1,rl

+ φ−1j Wpl−1,j,rlWpl−1,j,kWpl,j,kWpl,j,rl

=

√[2]9[3]3

[4]5λpl−1,ql−1,rlλpl,ql−1,kλpl−1,ql−1,kλpl,ql−1,rl

+

√[2]9[3]3

[4]5λpl−1,j,rlλpl,j,kλpl−1,j,kλpl,j,rl ,

so we have, for l = 1, 2, 3,(40)λpl−1,ql−1,rlλpl,ql−1,kλpl−1,ql−1,kλpl,ql−1,rl = −λpl−1,j,rlλpl,j,kλpl−1,j,kλpl,j,rl .

Then there are two solutions W+, W− for the cell system for E(12)2 .

Theorem 11.1. Every solution for the cells of E(12)2 is either equivalent to the

solution W+ or the inequivalent conjugate solution W−, given by

W±i,j,k =√

[2][3], W±pl,j,k =[2]√

[3]√[4]

,

W±pl,ql−1,rl=

√[2]

3

[4]

√[2]2 ±

√[2][4], W±pl,ql,rl+1

= −√

[2]3

[4]

√[2]2 ∓

√[2][4],

W±pl,ql,k = W±pl,j,rl+1=

√[2]

3

[4]

√[2][4]±

√[2][4],

W±pl,ql−1,k= W±pl,j,rl =

√[2]

3

[4]

√[2][4]∓

√[2][4],

for l = 1, 2, 3.

Proof. Let W ♯ be another solution for the cells of E(12)2 , which must be given

by W ♯v1,v2,v3 = λ♯v1,v2,v3 |W+

v1,v2,v3 | where the λ♯’s satisfy the condition (40).

We need to find unitaries uv1,v2 ∈ T, for v1, v2 vertices of E(12)2 , such that

upl,qluql,rl+1url+1,plλ

♯pl,ql,rl+1

= −1, l = 1, 2, 3, and uv1,v2uv2,v3uv3,v1λ♯v1,v2,v3 =

1 for all other triangles v1,v2,v3 on E(12)2 . We make the following choices:

uj,k = uk,i = uj,rl = uql,k = url+1,pl = 1, ui,j = λ♯i,j,k, upl,j = λ♯pl,j,rl+1,

uk,pl = λ♯pl,j,rl+1λ♯pl,j,k, url,pl = λ♯pl,j,rl+1

λ♯pl,j,rl , upl,ql = λ♯pl,j,kλ♯pl,ql,k

λ♯pl,j,rl+1,

upl,ql−1= λ♯pl,j,kλ

♯pl,ql−1,k

λ♯pl,j,rl+1, uql,rl+1

= −λ♯pl,j,rl+1λ♯pl,ql,kλ

♯pl,j,k

λ♯pl,j,rl+1,

for l = 1, 2, 3.Similarly, for any solution W ♯♯ with |W ♯♯

v1,v2,v3 | = |W−v1,v2,v3 |.The solutionsW+ andW− are not equivalent since |W+| 6= |W−|, and there

are no double edges on E(12)2 . We remark that the complex conjugate solutions

W± are equivalent to the solutions W∓: we choose a family of unitaries whichsatisfy (10) by uil,jl = ujl,kl = ukl,il = up,jl = ujl,r = uq,kl = ukl,p = 1,



uq,r = −1, and 2 × 2 unitary matrices uα = uβ = u where u is given byu(i, j) = 1− δi,j .

For E(12)2 , the cells W+ above give the following representation of the Hecke

algebra, where l = 1, 2, 3 (mod 3):

U (i,k) = U (j,i) = [2],

U (k,j) =i

pl

[2][3]

√[2]3

[3]√

[4]√[2]3

[3]√

[4]

[2]2

[3][4]

,

U (rl,j) =pl−1

pl

[2]2([2][4]+√

[2][4])

[3]2[4]

√[2]3√[3][4]√

[2]3√[3][4]

[2]2([2][4]−√

[2][4])

[3]2[4]

,

= U (k,ql) with rows labelled by p,pl+1,

U (ql,pl) =k

rl+1

[2][4]+√

[2][4]

[2][3]

−q

[2][4]−√

[2][4]

[2]√

[3]

−q

[2][4]−√

[2][4]

[2]√

[3]

[2]2−√

[2][4]

[2][3]

,

= U (pl,rl+1) with rows labelled by j, ql,

U (pl,rl) =j

ql−1

[2][4]−√

[2][4]

[2][3]

q[2][4]−

√[2][4]

[2]√

[3]q[2][4]−

√[2][4]

[2]√

[3]

[2]2+√

[2][4]

[2][3]

,

= U (ql−1,pl) with rows labelled by k, rl,

U (rl+1,ql) =pl

pl+1

[2]([2]2−√

[2][4])

[3]2−[2]√

[6]

−[2]√[6]

[2]([2]2+√

[2][4])

[3]2

,

U (pl,k) =

j

ql−1

ql

1[2]

q[2][4]−

√[2][4]√

[2][3][4]

q[2][4]+

√[2][4]√

[2][3][4]q[2][4]−

√[2][4]√

[2][3][4]

[2][4]−√

[2][4]

[3][4]

√[6]√

[3][4]q[2][4]+

√[2][4]√

[2][3][4]

√[6]√

[3][4]

[2][4]+√

[2][4]

[3][4]

.

12. E(12)1

For the graph E(12)1 (illustrated in Figure 14), we will use the notation

W(1)v1,v2,v3 for the cell of the triangle v1,v2,v3 where there are no double edges

between any of the vertices v1, v2, v3. For triangles that involve the doubleedges α, α′ or β, β′ we will specify which of the double edges is used by the



notation (ξ)v1,v2,v3 , and Wv1,v2,v3(ξ) := W (ξv1,v2,v3). Since the graph E(12)

1 is

a Z3-orbifold of the graph E(12)2 , we can obtain an orbifold solution for the

cells for E(12)1 as follows. We take the Z3-orbifold of E(12)

2 with the vertices i, jand k all fixed points- these are thus triplicated and become the vertices il, jland kl, l = 1, 2, 3, on E(12)

1 . The vertices p1, p2 and p3 on E(12)2 are identified

and become the vertex p on E(12)1 , and similarly the ql and rl become q and

r. The edges α1, α2 and α3 are identified and become the edge α on E(12)1 ,

also the edges α′1, α′2 and α′3 are identified and become the edge α′. Similarly

the edges βl, β′l and γl become the edges β, β′ and γ respectively on E(12)

1 .The Perron-Frobenius weights for the vertices are φil = 1, φjl = φkl = [3],l = 1, 2, 3, φp = [2][4] and φq = φr = [3][4]/[2]. Note that these are equal to

the Perron-Frobenius weights for the corresponding vertices of E(12)2 up to a

scalar factor of [4]/[2].

From the type I framesil• -

jl•, l = 1, 2, 3, we have |W (1)il,jl,kl

|2 = [2][3]

(which is equal to ([4]/[2])2|W (2)i,j,k|2/3). Then the type I frame

jl• -kl• ,

l = 1, 2, 3, gives |W (1)p,jl,kl

|2 = [3][4] (= ([4]/[2])2|W (2)pl,j,k

|2/3). Since the triangle

(α)p,jl,r

in E(12)1 comes from the triangle pl,j,rl in E(12)

2 , then

|W (1)p,jl,r(α)|2 =

[4]2

[2]2|W (2)

pl,j,rl|2 = [2]([2][4]∓

√[2][4]).

The triangle (α′)p,jl,r

in E(12)1 comes from the triangle pl,j,rl+1

in E(12)2 , giving

|W (1)p,jl,r(α′)|2 =

[4]2

[2]2|W (2)

pl,j,rl+1|2 = [2]([2][4]±

√[2][4]).

Similarly

|W (1)p,q,kl(β)|2 =

[4]2

[2]2|W (2)

pl,ql,k|2 = [2]([2][4]±

√[2][4]),

|W (1)p,q,kl(β′)|2 =

[4]2

[2]2|W (2)

pl,ql−1,k|2 = [2]([2][4]∓

√[2][4]).

The three triangles pl,ql,rl+1, l = 1, 2, 3, in E(12)

2 are identified in E(12)1 and

give the triangle (α′,β)p,q,r , so that |W (1)

p,q,r(α′,β)|2 = 3([4]/[2])2|W (2)pl,ql,rl+1 |2 =

([4]/[2])2([2]2 ∓√

[2][4]). Similarly |W (1)p,q,r(α,β′)|2 = 3([4]/[2])2|W (2)

pl,ql−1,rl |2 =

([4]/[2])2([2]2±√

[2][4]). Considering the type I frameq• -

r• gives the equa-

tion |W (1)p,q,r(α,β)|2+|W (1)

p,q,r(α,β′)|2+|W (1)p,q,r(α′,β)|2+|W (1)

p,q,r(α′,β′)|2 = [3]2[4]2/[2].

Substituting in for |W (1)p,q,r(α′,β)|2 and |W (1)

p,q,r(α,β′)|2 we find |W (1)p,q,r(α,β)|2 +

|W (1)p,q,r(α′,β′)|2 = 0, so that |W (1)

p,q,r(α,β)|2 = |W (1)p,q,r(α′,β′)|2 = 0. The reason for

this is that the triangle (α,β)p,q,r (and similarly for the triangle (α′,β′)

p,q,r ) in E(12)1



comes from the paths pl - ql - rl+1- pl+1 in E(12)

2 , which do notform a closed triangle.

From the type I framesr• -

-p• and

p• --

q•, we obtain the equations

λ1(α)λ1(α′) + λ2(α)λ2(α′) + λ3(α)λ3(α′) = 0,(41)

λ1(β)λ1(β′) + λ2(β)λ2(β′) + λ3(β)λ3(β′) = 0,(42)

where Wp,jl,r(ξ) = λl(ξ)|Wp,jl,r(ξ)|, for ξ ∈ α, α′, β, β′, l = 1, 2, 3. Anotherrestriction on the choice of phase is found by considering the type II framesjl• -

r• jm• , for l 6= m, Re(λl(α)λm(α′)λl(α′)λm(α)) = −1/2, and

similarly for the type II frameskl• -

p• km• , l 6= m, giving

λl(α)λm(α′)λl(α′)λm(α) = −1

2+ εl,m

√3

2i,(43)

λl(β)λm(β′)λl(β′)λm(β) = −1

2+ ε′l,m

√3

2i,(44)

where εl,m, ε′l,m ∈ ±1. Lastly, from the type II frame consisting of the

vertices jl, kl, q and r (l = 1, 2, 3) we have

(45) λl(α)λl(β′)λl(α′)λl(β) = −λ(αβ′)λ(α′β),

where Wp,q,r(ξ1,ξ2) = λ(ξ1,ξ2)|Wp,q,r(ξ1,ξ2)|, for ξ1 ∈ α, α′, ξ2 ∈ β, β′, l =1, 2, 3. Then for l 6= m,

λl(α)λm(α′)λl(α′)λm(α) = λl(β)λm(β′)λl(β′)λm(β),

and, from (43) and (44) we find εl,m = ε′l,m. Substituting in for λl(α)λl(α′)

from (43) into (41), we see that εl,l+1 = εm,m+1 for all l,m = 1, 2, 3, and thatεl,l−1 = −εl,l+1. Then the restrictions for the choice of phase are (45) and(46)

λl(α)λl+1(α′)λl(α′)λl+1(α) = λl(β)λl+1(β′)λl(β′)λl+1(β) = −1

2+ ε

√3

2i = eε

2πi3 ,

where ε ∈ ±1.Then we have obtained two orbifold solutions for the cell system for E(12)

1 :W+, W−.

Theorem 12.1. The following solutions W+, W− for the cells of E(12)1 are

inequivalent:

W±il,jl,kl =√

[2][3], W±p,jl,kl =√

[3][4],

W±p,jl,r(α) = ǫl√

[2]

√[2][4]±

√[2][4], W±p,jl,r(α′) = ǫl

√[2]

√[2][4]∓

√[2][4],

W±p,q,kl(β) = ǫl√

[2]

√[2][4]∓

√[2][4], W±p,q,kl(β′) = ǫl

√[2]

√[2][4]±

√[2][4],

W±p,q,r(αβ′) =[4]√[2]

√[2]2 ∓

√[2][4], W±p,q,r(α′β) = − [4]√

[2]

√[2]2 ±

√[2][4],

W±p,q,r(αβ) = W±p,q,r(α′β′) = 0,



for l = 1, 2, 3, where ǫ1 = 1 and ǫ2 = e2πi/3 = ǫ3.

Proof. The solutionsW+, W− are not equivalent, as can be seen by considering(10) for the trianglep,jl,r. We have the following two equations, for l = 1, 2, 3:

W+p,jl,r(α) = up,jlujl,r

(uα(α, α)W−p,jl ,r(α) + uα(α, α′)W−p,jl ,r(α′)

),

W+p,jl,r(α′) = up,jlujl,r

(uα(α′, α)W−p,jl,r(α) + uα(α′, α′)W−p,jl,r(α′)

).

So we require up,jl , ujl,r ∈ T and a 2 × 2 unitary matrix uα such that, forl = 1, 2, 3,

ǫl√

[2]x+ = up,jlujl,r

(uα(α, α)ǫl

√[2]x− + uα(α, α′)ǫl

√[2]x+

),(47)

ǫl√

[2]x− = up,jlujl,r

(uα(α′, α)ǫl

√[2]x− + uα(α′, α′)ǫl

√[2]x+

).(48)

where x± =√

[2][4]±√

[2][4]. Equation (47) must hold for each l = 1, 2, 3.

On the left hand side we have ǫl, hence we require uα(α, α′) = 0 becauseuα does not depend on l, and the difference in phase between ǫl and ǫl is 0,e−2πi/3, e2πi/3 respectively for l = 1, 2, 3 respectively. This difference in phasefor each l cannot come from up,jlujl,r (although up,jl , ujl,r do depend on l)

since in (48) the difference in phase is now 0, e2πi/3, e−2πi/3 respectively forl = 1, 2, 3 respectively, so we would need up,jlujl,r to take care of the phasedifference here, not up,jlujl,r. Then we have uα(α, α) = up,jlujl,r x+/x−, andsimilarly uα(α′, α) = 0 and uα(α′, α′) = up,jlujl,r x−/x+. But now uα is notunitary.

For E(12)1 , the cells W+ above give the following representation of the Hecke

algebra, where l = 1, 2, 3 (mod 3):

U (il,kl) = U (jl,il) = [2],

U (kl,jl) =il

p

[2][3]

√[2][4]

[3]√[2][4]

[3][4][3]

,

U (r,jl) =p(α)

p(α′)

[2]2([2][4]+√

[2][4])

[3]2[4]

ǫl√

[2]3√[3][4]

ǫl√

[2]3√[3][4]

[2]2([2][4]−√

[2][4])

[3]2[4]

= U (kl,q) with rows labelled by p(β′), p(β),

U (jl,p) =

kl

r(α)

r(α′)

1[2]

ǫl

q[2][4]+

√[2][4]√

[2][3][4]

ǫl

q[2][4]−

√[2][4]√

[2][3][4]

ǫl

q[2][4]+

√[2][4]√

[2][3][4]

[2][4]+√

[2][4]

[3][4]

ǫl√

[6]√[3][4]

ǫl

q[2][4]−

√[2][4]√

[2][3][4]

ǫl√

[6]√[3][4]

[2][4]−√

[2][4]

[3][4]

= U (p,kl) with rows labelled by jl, q(β′), q(β),



U (r,q) =

p(αβ)

p(αβ′)

p(α′β)

p(α′β′)

0 0 0 0

0[2]([2]2−

√[2][4])

[3]2 −√

[2]√[6]

0

0 −√

[2]√[6]

[2]([2]2+√

[2][4])

[3]2 0

0 0 0 0

,

U (p,r) with labels j1, j2, j3, q(β), q(β′)

=

[3][4] a− a+ −

√b+

√b−

a+[3][4] a− −ǫ2

√b+ ǫ2

√b−

a− a+[3][4] −ǫ2

√b+ ǫ2

√b−

−√b+ −ǫ2

√b+ −ǫ2

√b+

[2]2+√

[2][4]

[2][3] 0√b− ǫ2

√b− ǫ2

√b− 0

[2]2−√

[2][4]

[2][3]

= U (q,p) with labels k1, k3, k2, r(α′), r(α),

where a± = (−[2]2 ± i√

[2][4] )/[3][4], b± = ([2][4]±√

[2][4] )/[3][4]2.Our representation of the Hecke algebra is not equivalent to that given

by Sochen for E(12)1 in [41], however we believe that there is a typographical

error in Sochen’s presentation and that the weights he denotes by U (4,2r) =(U (3r ,6))∗ should be the complex conjugate of the one given. In this case,the representation of the Hecke algebra we give above can be shown to beequivalent by choosing a family of unitaries uil,jl = ujl,kl = ukl,il = up,jl =ukl,p = uq,r = 1, ujl,r = −ǫl = uq,kl and set the 2× 2 unitary matrices uα, uβto be the identity matrix.

13. E(12)5

We label the vertices of E(12)5 as in Figure 15. The Perron-Frobenius weights

associated to the vertices are φ1 = [3][6]/[2], φ2 = φ3 = φ8 = φ14 = [3][4]/[2],φ4 = φ5 = φ9 = φ15 = [3], φ6 = φ12 = [2][3]2/[6] = [2]2, φ7 = φ13 =[3]2[4]/[6] = [2][4], φ10 = φ16 = 1, φ11 = φ17 = [4]/[2]. The distinguished∗-vertex is vertex 10.

With Wv1,v2,v3 = λv1,v2,v3 |Wv1,v2,v3 |, λv1,v2,v3 ∈ T, we find two restrictionson the choice of phase

λ1,6,12λ2,7,12λ1,7,12λ2,6,12 = −λ1,6,13λ2,7,13λ1,7,13λ2,6,13,(49)

λ1,7,14λ1,8,13λ1,7,13λ1,8,14 = −λ3,7,14λ3,8,13λ3,7,13λ3,8,14.(50)

Theorem 13.1. There is up to equivalence a unique set of cells for E(12)5 given

by

W1,6,12 = W4,10,15 = W5,9,16 =√

[2][3],

W1,6,13 = W1,7,12 = [2]√

[3][4],



Figure 15. Labelled graph E(12)5

W1,7,13 = W3,7,14 = W3,8,13 = W3,8,17 = W3,11,14 = W2,7,15 = W2,9,13

= W4,7,14 = W5,8,13 =[4]√

[3]√[2]

,

W1,8,14 =[4]√

[3][6]√[2]

, W1,7,14 = W1,8,13 =

√[3][4][6]√

[2],

W2,6,12 = [4]√

[2], W2,6,13 = W2,7,12 = [2]√

[4],

W2,7,13 = −[4]√

[2], W3,7,13 = −[4]√

[6],

W3,8,14 =[4]√

[6]

[2], W4,7,15 = W5,9,13 =

√[3][4].

Proof. Let W ♯ be any other solution for the cells of E(12)5 . Then we have

W ♯v1,v2,v3 = λ♯v1,v2,v3 |Wv1,v2,v3 |, where the λ♯’s satisfy the conditions (49) and

(50). We need to find unitaries uv1,v2 ∈ T which satisfy u7,13u13,2u2,7λ♯2,7,13 =

−1, u7,13u13,3u3,7λ♯3,7,13 = −1 and uv1,v2uv2,v3uv3,v1λ

♯v1,v2,v3 = 1 for all other

trianglesv1,v2,v3 on E(12)5 . We set u2,7 = u2,9 = u3,8 = u3,11 = u6,13 = u7,13 =

u7,14 = u8,13 = u8,17 = u9,16 = u10,15 = u12,1 = u12,2 = u13,5 = u14,7 =

u15,2 = 1, u5,8 = λ♯5,8,13, u7,12 = λ♯2,7,12, u7,15 = λ♯2,7,15, u11,14 = −λ♯3,11,14,u13,1 = λ♯1,6,13, u13,2 = −λ♯2,7,13, u13,3 = λ♯3,8,13, u14,4 = λ♯4,7,14, u17,3 =

λ♯3,8,17, u1,7 = λ♯2,7,12λ♯1,7,12, u2,6 = −λ♯2,7,13λ♯2,6,13, u3,7 = −λ♯3,8,13λ♯3,7,13,

u9,13 = −λ♯2,7,13λ♯2,9,13, u15,4 = λ♯2,7,15λ♯4,7,15, u4,10 = λ♯4,7,15λ

♯2,7,15λ

♯4,10,15,

u5,9 = −λ♯2,9,13λ♯2,7,13λ♯5,9,13, u6,12 = −λ♯2,6,13λ♯2,6,12λ♯2,7,13,u14,1 = λ♯1,7,12λ

♯1,7,14λ

♯2,7,12, u14,3 = −λ♯3,7,13λ♯3,7,14λ♯3,8,13,



u1,6 = −λ♯2,6,12λ♯2,7,13λ♯1,6,12λ♯2,6,13, u1,8 = λ♯1,7,12λ♯1,7,13λ

♯1,8,13λ

♯2,7,12,

u8,14 = λ♯1,7,14λ♯1,8,13λ

♯1,7,13λ

♯1,8,14 and u16,5 = −λ♯2,7,13λ♯5,9,13λ♯2,9,13λ♯5,9,16.

For E(12)5 , we have the following representation of the Hecke algebra:

U (5,16) = U (16,9) = U (10,4) = U (15,10) = [2],

U (3,17) = U (17,8) = U (11,3) = U (14,11) =[2]

[4],

U (2,15) = U (4,14) = U (8,5) = U (9,2) =[4]

[3],

U (14,8) =3

1

1[2]

√[3]√[2]√

[3]√[2]

[3]

U (12,7) =2

1

1[2]

√[3]

[2]√[3]

[2][3][2]

= U (13,6) with rows labelled by 2, 1

U (3,13) =8

7

1[2] −

√[3]

[2]

−√

[3]

[2][3][2]

= U (7,3) with rows labelled by 14, 13

U (5,13) =9

8

1[2]

√[4]√[2]3√

[4]√[2]3

[4][2]2

= U (13,9) with labels 5,2

= U (7,4) with labels 15,14 = U (15,7) with labels 4,2

U (2,12) =7

6

[2][3]

√[2][4]

[3]√[2][4]

[3][4][3]



U (1,14) =7

8

[2][3]

[2]√

[4]

[3]

[2]√

[4]

[3][2][4][3]


U (12,6) =1

2

[3][2]3

[4]√

[3]

[2]3

[4]√

[3]

[2]3[4]2

[2]3

U (1,12) =6

7

1[6]

√[2][4]

[6]√[2][4]

[6][2][4][6]




U (13,8) =

5

3

1

1[2]

1[2]

√[6]

[2]√

[4]

1[2]

1[2]

√[6]

[2]√

[4]√[6]

[2]√

[4]

√[6]

[2]√

[4]

[6][2][4]

= U (14,7) with labels 4,3,1

U (3,14) =

8

7

11

1[2]

1√[3]

1√[3]

1√[3]

[2][3]

[2][3]

1√[3]

[2][3]

[2][3]

= U (8,3) with labels 14,13,17

U (2,13) =

9

7

6

1[2] − 1√

[3]

√[2]√

[3][4]

− 1√[3]

[2][3] −

√[2]3

[3]√

[4]√[2]√

[3][4]−√

[2]3

[3]√

[4]

[2]2

[3][4]

= U (7,2) with labels 15,13,12,

U (1,13) =

8

7

6

1[2]

√[4]

[2]√

[6]

√[2]3√[6]√

[4]

[2]√

[6]

[4][2][6]

√[2]3[4]

[6]√[2]3√[6]

√[2]3[4]

[6][2]2

[6]

= U (7,1) with labels 14,13,12

U (13,7) =

2

3

1

1[2]

√[6]√[2]3

−√

[3]

[2]2√[6]√[2]3

[6][2]2 −

√[3][6]√[2]5

−√

[3]

[2]2 −√

[3][6]√[2]5

[3][2]3

.

14. E(24)

We label the vertices of the graph E(24) as in Figure 16. The Perron-Frobenius weights are: φ1 = φ8 = 1, φ2 = φ7 = [2][4], φ3 = φ6 = [4][5]/[2],φ4 = φ5 = [4][7]/[2], φ9 = φ16 = φ17 = φ24 = [3], φ10 = φ15 = φ18 = φ23 =[3][4]/[2], φ11 = φ14 = φ19 = φ22 = [3][5] and φ12 = φ13 = φ20 = φ21 = [9].With [a] = [a]q, q = eiπ/24, we have the relation [4]2 = [2][10].

The following cells follow from the A case: |W1,9,17|2 = |W8,16,24|2 = [2][3],|W2,9,17|2 = |W7,16,24|2 = [3][4], |W2,9,18|2 = |W2,10,17|2 = |W7,15,24|2 =|W7,16,23|2 = [3]2[4], |W2,10,19|2 = |W2,11,18|2 = |W7,14,23|2 = |W7,15,22|2 =[3][4][5], |W2,11,19|2 = |W7,14,22|2 = [3]2[4][5] and |W3,10,19|2 = |W3,14,23|2 =|W6,11,18|2 = |W6,15,22|2 = [3][4]2[5]/[2].



Figure 16. Labelled graph E(24)

The type II frame2• -

19• 4• gives φ−1

11 |W2,11,19|2|W4,11,19|2 =[3][4]2[5][7], and so we obtain |W4,11,19|2 = [4][5][7]. From the type I frame11• -

19• we have the equation |W2,11,19|2 + |W4,11,19|2 + |W5,11,19|2 =[2][3]2[5]2, giving |W5,11,19|2 = [4][5][7] = |W4,11,19|2. Then by considering the

type I frames4• -

11• and22• -

4•, we find |W4,14,22|2 = |W5,14,22|2 =|W4,11,19|2 = |W5,11,19|2. Similarly |W4,12,19|2 = |W4,14,21|2 = |W5,11,20|2 =|W5,13,22|2 and |W3,12,19|2 = |W3,14,21|2 = |W6,11,20|2 = |W6,13,22|2, and thecells have a Z2 symmetry.

From type I frames we have the equations:

|W4,11,19|2 + |W4,12,19|2 + |W4,14,19|2 = [3][4][5][7],(51)

|W3,12,19|2 + |W4,12,19|2 = [2][3][5][9],(52)

|W3,10,19|2 + |W3,12,19|2 + |W3,14,19|2 = [3][4][5]2,(53)

|W3,14,19|2 + |W4,14,19|2 + |W5,14,19|2 = [2][3]2[5]2,(54)

|W3,12,19|2 + |W3,12,21|2 = [4][5][9],(55)

|W3,12,21|2 + |W4,12,21|2 = [2][9]2.(56)

The type II frame11• -

19• 12• , gives φ−1

4 |W4,11,19|2|W4,12,19|2 =[3]2[5]2[9], so |W4,12,19|2 = [3]2[5][9]/[2]. Then using the equations (51)-(56)we obtain |W4,14,19|2 = [5]2[7]/[2], |W3,12,19|2 = [3][5][9]/[2], |W3,14,19|2 =[3]2[5]2/[2], |W5,14,19|2 = [5][7][10], |W3,12,21|2 = [5]2[9]/[2] and |W4,12,21|2 =[7][9]/[2].

With Wv1,v2,v3 = λv1,v2,v3 |Wv1,v2,v3 |, λv1,v2,v3 ∈ T, we have the followingrestrictions on the λ’s:

λ3,12,19λ3,14,21λ3,12,21λ3,14,19 = −λ4,12,19λ4,14,21λ4,12,21λ4,14,19,(57)

λ4,11,22λ4,14,19λ4,11,19λ4,14,22 = −λ5,11,22λ5,14,19λ5,11,19λ5,14,22,(58)



λ5,11,20λ5,13,22λ5,11,22λ5,13,20 = −λ6,11,20λ6,13,22λ6,11,22λ6,13,20.(59)

Theorem 14.1. There is up to equivalence a unique set of cells for E(24) givenby

W1,9,17 = W8,16,24 =√

[2][3], W2,9,17 = W7,16,24 =√

[3][4],

W2,9,18 = W2,10,17 = W7,15,24 = W7,16,23 = [3]√

[4],

W2,10,19 = W2,11,18 = W7,14,23 = W7,15,22 =√

[3][4][5],

W2,11,19 = W7,14,22 = [3]√

[4][5],

W3,10,19 = W3,14,23 = W6,11,18 = W6,15,22 =[4]√

[3][5]√[2]

,

W4,11,19 = W4,14,22 = W5,11,19 = W5,14,22 =√

[4][5][7],

W4,12,19 = W4,14,21 = W5,11,20 = W5,13,22 =[3]√

[5][9]√[2]

,

W3,12,19 = W3,14,21 = W6,11,20 = W6,13,22 =

√[3][5][9]√

[2],

W3,14,19 = W6,11,22 =[3][5]√

[2], W4,14,19 = W5,11,22 =

[5]√

[7]√[2]

,

W5,14,19 =√

[5][7][10], W4,11,22 = −√

[5][7][10],

W3,12,21 = W6,13,20 = − [5]√

[9]√[2]

, W4,12,21 = W5,13,20 =

√[7][9]√[2]

.

Proof. Let W ♯ be any solution for the cells of E(24). Then W ♯v1,v2,v3 =

λ♯v1,v2,v3 |Wv1,v2,v3 |, where the λ♯’s satisfy the conditions (57), (58) and (59).

We need to find unitaries uv1,v2 ∈ T such that u12,21u21,3u3,12λ♯3,12,21 = −1,

u13,20u20,6u6,13λ♯6,13,20 = −1, u11,22u22,4u4,11λ

♯4,11,22 = −1, and for all other

triangles v1,v2,v3 on E(24) we require uv1,v2uv2,v3uv3,v1λ♯v1,v2,v3 = 1. We make

the following choices for the uv1,v2 :

u3,12 = u3,14 = u4,11 = u5,13 = u5,14 = u11,20

= u14,19 = u20,6 = u21,3 = u21,4 = u22,6 = 1,

u12,21 = −λ♯3,12,21, u14,21 = λ♯3,14,21, u19,3 = λ♯3,14,19, u19,5 = −λ♯5,14,19,

u4,12 = −λ♯3,12,21λ♯4,12,21, u4,14 = λ♯3,14,21λ♯4,14,21, u6,11 = λ♯5,14,22λ

♯6,11,20,

u12,19 = λ♯3,14,19λ♯3,12,19, u11,22 = λ♯6,11,20λ

♯5,14,22λ

♯6,11,22,

u19,4 = λ♯4,14,21λ♯3,14,21λ

♯4,14,19, u22,4 = −λ♯5,14,22λ♯6,11,22λ♯4,11,22λ♯6,11,20,

u5,11 = −λ♯4,11,22λ♯4,14,21λ♯5,14,22λ♯3,14,21λ♯4,14,22λ♯5,11,22,

u11,19 = −λ♯3,12,21λ♯3,14,19λ♯4,12,19λ♯3,12,19λ♯4,11,19λ♯4,12,21,



u20,5 = −λ♯3,14,21λ♯4,14,22λ♯5,11,22λ♯4,11,22λ♯4,14,21λ♯5,11,20,

u22,5 = −λ♯3,14,21λ♯4,14,22λ♯6,11,22λ♯4,11,22λ♯4,14,21λ♯6,11,20,

u13,22 = −λ♯4,11,22λ♯4,14,21λ♯6,11,20λ♯3,14,21λ♯4,14,22λ♯5,13,22λ♯6,11,22,

u14,22 = −λ♯4,11,22λ♯4,14,21λ♯6,11,20λ♯3,14,21λ♯4,14,22λ♯5,14,22λ♯6,11,22,

u6,13 = −λ♯3,14,21λ♯4,14,22λ♯5,13,22λ♯6,11,22λ♯4,11,22λ♯4,14,21λ♯6,11,20λ♯6,13,22,

u13,20 = λ♯4,11,22λ♯4,14,21λ

♯6,11,20λ

♯6,13,22λ

♯3,14,21λ

♯4,14,22λ

♯5,13,22λ

♯6,11,22λ

♯6,13,20.

The uv1,v2 involving the vertices 1, 2, 7, 8, 9, 10, 15, 16, 17, 18, 23 and 24are chosen in the same way as in the proof of uniqueness of the cells for the Agraphs.

For E(24), we have the following representation of the Hecke algebra (weomit those weights which come from the A(24) graph):

U (3,21) =12

14

[5][4] −

√[3][5]

[4]

−√

[3][5]

[4][3][4]

= U (12,3) with labels 21,19


U (19,12) =3

4

1[2]

√[3]

[2]√[3]

[2][3][2]


= U (20,11) with labels 6,5 = U (22,13) with labels 6,5,

U (5,19) =11

14

[2][3]

√[2][4]

[3]√[2][4]

[3][4][3]

= U (14,5) with labels 22,19

U (4,22) =14

11

[2][3] −

√[2][4]

[3]

−√

[2][4]

[3][4][3]

= U (11,4) with labels 19,22,

U (20,13) =6

5

[5]2

[2][9] − [5]√

[7]

[2][9]

− [5]√

[7]

[2][9][7]

[2][9]


U (4,21) =12

14

1[4]

[3]√

[5]

[4]√

[7]

[3]√

[5]

[4]√

[7]

[3]2[5][4][7]

= U (12,4) with labels 21,19




U (19,14) =

3

4

5

1[2]

√[7]

[2][3]

√[7][10]

[3]√

[2][5]√[7]

[2][3][7]

[2][3]2[7]√

[10]

[3]2√

[2][5]√[7][10]

[3]√

[2][5]

[7]√

[10]

[3]2√

[2][5]

[7][10][3]2[5]

U (22,11) =

6

5

4

1[2]

√[7]

[2][3] −√

[7][10]

[3]√

[2][5]√[7]

[2][3][7]

[2][3]2 − [7]√

[10]

[3]2√

[2][5]

−√

[7][10]

[3]√

[2][5]− [7]

√[10]

[3]2√

[2][5]

[7][10][3]2[5]

U (19,11) =

2

4

5

[4][5]

[4]√

[7]

[3][5]

[4]√

[7]

[3][5]

[4]√

[7]

[3][5][4][7][3]2[5]

[4][7][3]2[5]

[4]√

[7]

[3][5][4][7][3]2[5]

[4][7][3]2[5]

= U (22,14) with labels 7,4,5

U (3,19) =

10

14

12

[4][5]

√[3]√[5]

√[9]

[5]√[3]√[5]

[3][4]

√[3][9]

[4]√

[5]√[9]

[5]

√[3][9]

[4]√

[5]

[9][4][5]

= U (14,3) with labels 23,19,21

= U (6,22) with labels 15,11,13 = U (11,6) with labels 18,22,20

U (4,19) =

11

14

12

[2][3]

√[2][5]

[3]√

[4]

√[2][9]√[4][7]√

[2][5]

[3]√

[4]

[5][3][4]

√[5][9]

[4]√

[7]√[2][9]√[4][7]

√[5][9]

[4]√

[7]

[3][9][4][7]

= U (14,4) with labels 22,19,21 = U (5,22) with labels 14,11,13

= U (11,5) with labels 19,22,20.

The Hecke representation given above cannot be equivalent to that given bySochen in [41] for E(24) as our weights [U (14,4)]19,19, [U (14,4)]21,21, [U (11,5)]20,20,

[U (11,5)]22,22 and [U (19,11)]2,2 (as well as the corresponding weights under thereflection of the graph which sends vertices 1 - 8) have different absolutevalues to those given by Sochen (and there are no double edges on the graph).We do not believe that there exists two inequivalent solutions for the Heckerepresentation for E(24), and that the differences must be due to typographicalerrors in [41].

Acknowledgements. This paper is based on work in [40]. The first author waspartially supported by the EU-NCG network in Non-Commutative Geometry



MRTN-CT-2006-031962, and the second author was supported by a scholarshipfrom the School of Mathematics, Cardiff University.

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Received February 2, 2009; accepted May 25, 2009

David E. Evans and Mathew PughSchool of Mathematics, Cardiff UniversitySenghennydd Road, Cardiff, CF24 4AG, Wales, UKE-mail: evansde,[email protected]




An enrichment of KK -theory over the category

of symmetric spectra

Michael Joachim and Stephan Stolz



Abstract. In [6] Higson showed that the formal properties of the Kasparov KK -theorygroups are best understood if one regards KK (A, B) for separable C∗-algebras A, B as themorphism set of a category KK . In category language the composition and exterior KK -product give KK the structure of a symmetric monoidal category which is enriched overabelian groups. We show that the enrichment of KK can be lifted to an enrichment over thecategory of symmetric spectra.

1. Introduction

A fundamental tool in Index theory and in the theory of C∗-algebras isKasparov’s bivariantK-theory which associates to C∗-algebrasA, B an abeliangroup KK (A,B) which is contravariant in A and covariant in B. Central toKasparov’s theory is the construction of a product

(1.1) KK (A1, C1 ⊗B)⊗KK (A2 ⊗B,C2)→ KK (A1 ⊗A2, C1 ⊗ C2),

which contains and generalizes a number of constructions from K-theory andindex theory. For example, if π is a discrete group with classifying space Bπand reduced C∗-algebra C∗π, and we set A1 = A2 = R, B = C0(Bπ), C1 =C∗π, C2 = C0(Rn), then the Kasparov product with the so-called Mischenko-Fomenko element ν ∈ KK (R, C∗π ⊗ C0(Bπ)) gives a homomorphism

(1.2) KOn(Bπ) = KK (C0(Bπ), C0(Rn))→KK (R, C∗π ⊗ C0(Rn)) = KOn(C

∗π),

which is known as the assembly map in (real) K-theory.As Higson explains very nicely in [6], the plethora of formal properties of

the Kasparov product (1.1) is best organized by thinking of an element of

The second author was supported by NSF grants DMS 0707068 and DMS 0757253.

144 Michael Joachim and Stephan Stolz

KK (A,B) as a “generalized” ∗-homomorphism from A to B. Specializing theKasparov product (1.1), there is a product

(1.3) KK (A,B) ⊗KK (B,C) −→ KK (A,C),

to be thought of as “composition” of generalized homomorphisms, and a prod-uct

(1.4) KK (A1, A2)⊗KK (C1, C2) −→ KK (A1 ⊗ C1, A2 ⊗ C2)

that is interpreted as “tensor product” of generalized homomorphisms. Thegeneral Kasparov product (1.1) of f ∈ KK (A1, C1⊗B) and g ∈ KK (A2⊗B,C2)is then given by the composition

A1 ⊗A2f⊗1−→ C1 ⊗B ⊗A2

1⊗g−→ C1 ⊗ C2.

In particular, the general Kasparov product (1.1) can be expressed in terms ofthe composition product (1.3) and the external product (1.4). All the formalproperties of these products can be nicely expressed by saying that there is acategory KK with the following properties

• the objects of KK are the separable C∗-algebras• the morphisms from A to B form the abelian group KK (A,B), and

the composition of morphisms is given by the product (1.3)• KK is a symmetric monoidal category; the tensor product for objects

is given by the (spatial) tensor product of C∗-algebras; the tensorproduct for morphisms is given by the product (1.4).• there is a functor C : C∗ → KK from the category of (separable) C∗-

algebras to KK which is the identity on objects and which is compatiblewith the symmetric monoidal structure on both categories.

We note that the functor C gives for C∗-algebras A, B a map C∗(A,B) →KK (A,B), where C∗(A,B) is the space of morphisms from A to B in thecategory C∗ consisting of all ∗-homomorphisms from A to B. This map can bethought of as associating a “generalized” homomorphism to each ∗-homomor-phism in a way that is compatible with composition product and externalproduct.

We observe that KK is a category enriched over the category of abeliangroups (also called an preadditive category) in the sense that the morphismsKK (A,B) form an abelian group and that the composition law (1.3) is ahomomorphism of abelian groups. The formal properties of the compositionand external products can be expressed in the lingo of category theory bysaying that KK is a symmetric monoidal preadditive category.

The main result of this paper is that the above statement can be “spec-trified” in the following sense. Let K = K(F) be the (real resp. complex)K-theory spectrum, which is a commutative ring spectrum in the world ofsymmetric spectra (see Section 7). As in the category of modules over a com-mutative ring, there is a product M ∧K N of K-module spectra M , N , whichis again a K-module spectrum. This “smash” product over K gives the cat-egory K -Mod of K-module spectra the structure of a a symmetric monoidal


An enrichment of KK -theory 145

category. Associating to a K-module spectrum X its homotopy group π0(X)gives a functor

π0 : K -Mod −→ Ab

to the category of abelian groups which is compatible with the symmetricmonoidal structure on these categories.

Theorem 1.5. There is a symmetric monoidal category KK enriched overthe category of K-module spectra such that the symmetric monoidal preadditivecategory obtained from KK by applying the functor π0 is KK.

This result can be expressed in a less technical, but also less precise form inthe following way.

Theorem 1.6. For separable, Z/2-graded C∗-algebras A and B there is aK-module spectrum KK(A,B) and there are maps

c : KK(A,B) ∧K KK(B,C) −→ KK(A,C)(1.7)

m : KK(A1, A2) ∧K KK(C1, C2) −→ KK(A1 ⊗ A2, C1 ⊗ C2),(1.8)

with the following properties:

(1) there are isomorphisms

π0KK(A,B) ∼= KK (A,B)

for C∗-algebras A, B.(2) the map (1.7) induces on π0 the composition product (1.3).(3) the map (1.8) induces on π0 the external product (1.4).(4) The products (1.7) and (1.8) satisfy a collection of (quite natural) as-

sociativity and compatibility conditions (which are spelled out in thedefinition of an enriched symmetric monoidal category in Section 6).

Here π0KK(A,B) is the zeroth homotopy group of the symmetric spectrumKK(A,B) (cp. Definition 7.8). Applying a criterion of Hovey, Shipley andSmith (the second part of their Proposition 5.6.4), we see that KK(A,B) isa semistable symmetric spectrum, which in turn implies that π0(KK(A,B))can be identified with [S,KK(A,B)], the group of homotopy classes of spec-trum maps from the sphere spectrum to KK(A,B) (the morphisms from Sto KK(A,B) in the associated “homotopy category”). Since KK(A,B) is aK-module spectrum, we may identify [S,KK(A,B)] with the group [K, X ]K ofhomotopy classes of K-module maps. In particular, the composition productmap c of (1.7) induces a homomorphism of abelian groups

[K,KK(A,B)]K ⊗ [K,KK(B,C)]K −→ [K,KK(A,C)]K

given by sending the tensor product of maps f : K −→ KK(A,B) and g : K −→KK(B,C) to the composition

K = K ∧K Kf∧g−→ KK(A,B) ∧K KK(B,C)

c−→ KK(A,C).

The claim in (2) is that this pairing is equal to the composition pairing (1.3) viathe identification [K,KK(A,B)]K = π0(KK(A,B)) = KK (A,B) (the second



equality is provided by part (1) of the theorem). Part (3) of the theorem iscompletely analogous.

We like to think of this result as a “dictionary” that allows to translatebetween KK -groups—the central objects in index theory and the theory ofC∗-algebras—and maps between spectra, the primary objects in stable homo-topy theory. We want to illustrate this in two simple examples.

We note that for any (separable) C∗-algebra B we have the K-module spec-

trum KB def= KK(F, B), whose homotopy group πn(KB) is the K-theory group

Kn(B). Taking the adjoint of the map c in the special case A = F, we obtaina K-module map

KK(B,C) −→ HomK(KB,KC).

The associativity properties of c imply that this map provides us with a functor

KK −→ K -Mod

of categories enriched over the category of K-module spectra; here the objectsof K -Mod are the K-module spectra, and the morphisms from X to Y isthe K-module spectrum HomK(X,Y ). Moreover, this functor is a functor ofsymmetric monoidal categories, where the monoidal structure in K -Mod isgiven by the smash product over K. Passing to the homotopy category, thefunctor gives on morphisms a homomorphism

KK (B,C) = π0(KK (B,C)) −→ π0(HomK(KB,KC)) = [KB,KC]K,

which is compatible with the composition and the external product on bothsides (such a map compatible with composition was constructed before bySchlichting [10, 15]). Summarizing, we see that this construction producesfrom KK -elements (the kind of objects index theory people play with) mapsbetween symmetric module spectra (the toys of stable homotopy theorists);moreover, the composition resp. external product of KK -elements correspondsto the composition resp. smash product of module maps.

This shows that any map between K-theory groups that has been producedby constructing certain KK -elements is induced by a map of the correspondingK-theory spectra, simply by replacing all KK -elements by the correspondingmaps between spectra, and all Kasparov products by the appropriate compos-tions/smash products of these maps. To illustrate this, let us construct themap of K-theory spectra inducing the assembly map (1.2) (of course the factthat the assembly map comes from a map of spectra is well-known).

We recall that the assembly map is given by the Kasparov product withthe Fomenko-Mischenko element ν ∈ KK (R, C∗π ⊗ C0(Bπ)) = [K,K(C∗π ⊗C0(Bπ))]K (assume that Bπ is compact for simplicity). Identifying ν with (thehomotopy class of) a K-module map we obtain a map of K-module spectra

K ∧Bπ+≃−→ K ∧K KK(C0(Bπ+),R) −→

KK(R, C0(Bπ+)⊗ C∗π) ∧K KK(C0(Bπ+)⊗ C∗π,C∗π)c→ K(C∗π)



whose induced map on homotopy groups

A∗ : πn(K ∧Bπ+) = KOn(Bπ)→ πn(K(C∗π)) = KOn(C∗π)

is the assembly map. One advantage of this spectrum level version of the as-sembly map is that it allows us to “introduce coefficients”; this is an importantmove, since it turns out that for a finite group π the assembly map is essentiallytrivial, while it is an isomorphism with coefficients in Q/Z.

It should be emphasized that the point of the paper is a translation betweenthe Kasparov product on the one hand and the composition/smash productof maps between K-module spectra on the other hand; in particular, no newoperator theoretic statement is obtained—except a generalization of Higson’saxiomatic characterization of the Kasparov product to the case of Z/2-gradedC∗-algebras.

This paper is organized as follows. In Section 2 we describe the elements ofKK (A,B) following Cuntz as “generalized” ∗-homomorphisms from A to B.In Section 3 we review Higson’s axiomatic characterization of the KK -groups.This leads in particular to a uniqueness statement concerning the compositionand tensor product of KK -elements. In Sections 4 (resp. 5) we describe thecomposition (resp. tensor product) of KK -elements in the Cuntz picture. Thedefinition of an “enriched” category is reviewed in Section 6 (for objects A,B in an “enriched” category D, the morphisms D(A,B) from A to B formnot just a set, but have more structure: D(A,B) could be a topological space,an abelian group, or—more generally—an object in a symmetric monoidalcategory). In Section 7 we define symmetric spaces and spectra. In Sections 8and 9 we describe a recipe to produce (symmetric monoidal) categories whichare enriched over the category of symmetric spaces (or spectra). In Section 10we apply this recipe to produce the category KK (which is enriched over thecategory of symmetric spectra meaning that for objects A, B in this category,the morphisms from A to B constitute a symmetric spectrum). Finally, inSection 11, we extend these results to Z/2-graded C∗-algebras.

For simplicity we shall assume that all C∗-algebras considered in the follow-ing are separable.

2. The Cuntz picture of KK -theory

There are basically three descriptions of the abelian groups KK (A,B):

• The original definition of Kasparov [8], where KK (A,B) is defined asthe set of equivalence classes of “Kasparov A−B-bimodules”;• The “Cuntz picture”, where KK (A,B) consists of homotopy classes of

“quasi-homomorphisms” from A to B;• Higson’s axiomatic characterization of KK (A,B).

This paper is based on the Cuntz picture of KK -theory; basically the spectrumKK(A,B) is build from spaces of quasi-homomorphisms. As a fairly directconsequence of the construction we obtain an isomorphism of sets

π0KK(A,B)←→ KK (A,B).



Then it will be convenient for us to use Higson’s axiomatic characterizationas a tool to check that our map is compatible with the group structure, thecomposition product and the external product on both sides.

2.1. Quasi-homomorphisms. Let A,B be C∗-algebras. A quasi-homomor-phism from A to B is a ∗-homomorphism

f : qA −→ K⊗B.Here

• qA is the Cuntz algebra, a C∗-algebra functorially associated to Adefined below, and• K is the C∗-algebra of compact operators on a fixed (separable) Hilbert

space H , and K⊗B is the (spatial) tensor product of C∗-algebras [11,Chapter T].

We note that there is a “stabilization map”

(2.2) C∗(A,B) −→ C∗(qA,K ⊗B) f 7→ (e⊗ 1B) f π0

from ∗-homomorphisms to quasi-homomorphisms. Here π0 : qA → A is a∗-homomorphism defined in 2.5 below, e : F → K is the ∗-homomorphismwhich sends the unit to a fixed rank one projection operator, 1B is the identityon B, and e⊗ 1B : K = F⊗B → K⊗B is their tensor product.

In the “Cuntz picture” KK (A,B) is defined as

(2.3) KK (A,B)def= [qA,K ⊗B],

where [qA,K⊗B] denotes the homotopy classes of ∗-homomorphisms from qAto K ⊗B; in other words, KK (A,B) is defined as the set of homotopy classesof quasi-homomorphisms from A to B.

The Cuntz algebra is an ideal in the free product A ∗A of two copies of A.Before defining the Cuntz algebra, we will recall the construction of the freeproduct of C∗-algebras.

2.4. Free product of C∗-algebras. Let Ai, i ∈ I be a family of C∗-algebras.Then the free product ∗

i∈IAi is a C∗-algebra which is the coproduct of the Ai’s

in the category ofC∗-algebras; i.e., there are ∗-homomorphisms ιi : Ai → ∗i∈IAi,

such that for any C∗-algebra B the map

C∗( ∗i∈IAi, B)→

∏

i∈I

C∗(Ai, B) f 7→ f ιi

is a bijection. The construction of the free product ∗i∈IAi is reminiscent of

the construction of the free product of groups and goes as follows. Consider“words”

a1a2 . . . am ai ∈⋃

i∈I

Ai,

whose “letters” a1, . . . , am are elements of the Ais. Here we identify a word

a1 . . . ak−1akak+1ak+2 . . . am with a1 . . . ak−1aak+2 . . . am



if ak and ak+1 belong to the same algebra Ai, and a = ak · ak+1 ∈ Ai. Wedefine the algebraic free product ∗

i∈I

algAi to be the vector space of finite linear

combinations of such words. This is a ∗-algebra with multiplication given byconcatenation of words and anti-involution ∗ given by (a1 . . . an)

∗ = a∗n . . . a∗1.

The free product ∗i∈IAi is the completion of ∗

i∈I

algAi with respect to the maximal

C∗-norm||z||max = sup

π||π(z)||

Here the supremum is taken over all ∗-homomorphisms π : ∗i∈I

alg Ai → B(H)

to the C∗-algebra of bounded operators on some Hilbert space H (this is finitesince ||a1 . . . an||max ≤ ||a1|| · · · · · ||an||).2.5. The Cuntz algebra. The Cuntz algebra qA associated to a C∗-algebra

A is an ideal in the free product QAdef= A ∗A of two copies of A. To describe

elements in QA it is convenient to write QA = A1 ∗A2, where the superscriptsare used to distinguish the two copies of A in QA. In particular for eacha ∈ A, there are two 1-letter words, namely a1 and a2, where the superscriptindicates which copy of A the letter a comes from. The Cuntz algebra is theclosed two-sided ideal in QA generated by the elements

q(a)def= a1 − a2 a ∈ A.

Let π0 be the C∗-homomorphism

π0 : QA→ A a1 7→ a, a2 7→ 0.

Abusing notation, we will also write π0 : qA → A for the restriction of π0 toqA ⊂ QA.

2.6. The group structure on KK (A,B). A choice of an isomorphism H ⊕H ∼= H determines a C∗-homomorphism K ⊕ K → K, which is then used todefine an “addition”.

+: KK (A,B)× KK (A,B) −→ KK (A,B),

where f1 + f2 is defined to be the composition

qAf1⊕f2−→ (K ⊗B)⊕ (K ⊗B) = (K ⊕K) ⊗B −→ K⊗B.

This gives KK (A,B) the structure of an abelian group. Inverses are obtainedby precomposition with the natural transposition on qA which is induced byinterchanging the two copies of A in QA (cp. [3, p. 37]).

3. Higson’s axiomatic characterization of KK -theory

3.1. Properties of KK (A,B). The abelian groups KK (A,B) have a numberof functorial properties; in particular, when considered in conjunction withthe composition product (1.3) and the external product (1.4). Fortunately,it turns out that all the other properties can be recovered from the followingthree “basic” properties.



Let us fix a C∗-algebra A. Then we can consider B 7→ KK (A,B) as acovariant functor

Fdef= KK (A,−) : C∗ −→ Ab

from the category of C∗-algebras to the category of abelian groups. Thisfunctor has the following three properties:

(i) (Homotopy Invariance) If the ∗-homomorphisms f, f ′ : B → B′ arehomotopic, then F (f) = F (f ′).

(ii) (Stability) Let K be the C∗-algebra of compact operators, and lete : F→ K be the ∗-homomorphism induced by the choice of a rank oneprojection. Then for any C∗-algebra B the induced homomorphismF (e⊗ 1B) : F (B)→ F (K ⊗B) is an isomorphism.

(iii) (Split Exactness) The functor F applied to a split exact sequence ofC∗-algebras gives a split exact sequence (of abelian groups).

We note that the first two properties of F (B) = [qA,K⊗B] follow quite directlyfrom the definition.

The following result of Higson gives an axiomatic characterization of theabelian groups KK (A,B).

Theorem 3.2 ([6]). Given a functor F from C∗ to the category of abeliangroups with the above properties, and an element x ∈ F (A), then there existsa unique natural transformation α : KK (A,−)→ F such that αA(1A) = x.

We recall that a natural transformation α : KK (A,−)→ F consists of a col-lection of homomorphisms αB : KK (A,B)→ F (B), one for each C∗-algebra Bwhich are compatible with induced maps in the sense that for every ∗-homomor-phism f : B → C the following diagram commutes:

KK (A,B)αB−−−−→ F (B)

f∗

yyf∗

KK (A,C)αC−−−−→ F (C)

3.3. Addendum. The uniqueness statement in the above theorem can bestrengthened: any natural transformation α : KK (A, ) → F between thesefunctors considered as functors with values in the category of sets (i.e., theαBs are not required to be group homomorphisms) is automatically a naturaltransformation of groups. This is a byproduct of the proof of Theorem 3.2.

Higson proved Theorem 3.2 using the Kasparov definition of KK (A,B) [6,Theorem 4.8]. Using the fact that the Cuntz groups [qA,K⊗B] are isomorphicto the Kasparov groups (via a natural transformation), this of course impliesthe result above. However, since a direct proof is fairly straightforward andmight shed a light on the construction of the Cuntz algebra qA, we will provetheorem 3.2.



Proof of Theorem 3.2. The main ingredient of the proof is the result due toCuntz [3, Prop. 3.1(b)] that for any homotopy invariant, stable, split-exactfunctor

F : C∗ → Ab

the induced map(π0)∗ : F (qA)→ F (A)

is an isomorphism. Suppose that α : KK (A,−) → F is a natural transforma-tion (of set-valued functors) with αA(1A) = x ∈ F (A). Let f : qA → K ⊗ Bbe a ∗-homomorphism and let [f ] ∈ [qA,K⊗B] = KK (A,B) be its homotopyclass. To show that α([f ]) ∈ F (B) is determined by x consider the followingcommutative diagram

KK (A,A)αA−−−−→ F (A)

(π0)∗

x∼= (π0)∗

x∼=

KK (A, qA)αqA−−−−→ F (qA)

f∗

y f∗

y

KK (A,K ⊗B)αK⊗B−−−−→ F (K ⊗B)

(e⊗1)∗

x∼= (e⊗1)∗

x∼=

KK (A,B)αB−−−−→ F (B)

It is easy to check that [f ] ∈ KK (A,B) is the image of [1A] ∈ KK (A,A) underthe composition of the vertical homomorphisms on the left. Hence αB([f ])equals the image of x ∈ F (A) under the composition of the vertical homomor-phisms on the right. In particular, αB is determined by x ∈ F (B) via theformula

αB([f ]) = (e⊗ 1)−1∗ f∗ (π0)

−1∗ (x).

4. KK as a category

Following Cuntz [3] we will describe in this section how to “compose” (homo-topy classes of) quasi-homomorphisms to obtain an associative bilinear product

(4.1) : KK (A,B) ×KK (B,C)→ KK (A,C).

This extends the usual composition of ∗-homomorphisms in the sense that thenatural map

C : C∗(A,B)→ KK (A,B)

from ∗-homomorphisms to quasi-homomorphisms sends the composition f gof two ∗-homomorphisms to the composition C(f) C(g) of the correspondingquasi-homomorphisms. In other words, C is a functor

C : C∗ → KK ,

where



• the objects of the category C∗ are the C∗-algebras, and the morphismsfrom A to B are the ∗-homomorphisms;• the objects of the category KK are the C∗-algebras, and the morphisms

from A to B are (homotopy classes of) quasi-homomorphisms, i.e.,elements of KK (A,B).

For our purposes of defining a composition of (homotopy classes of) quasi-homomorphisms it is convenient to replace [qA,K⊗B] by limn[q

nA,Kn ⊗B],where Kn = K ⊗ · · · ⊗ K is the tensor product of n copies of K, and the limitis taken with respect to the stabilization homomorphism

(4.2) [qnA,Kn ⊗B] −→ [qn+1A,Kn+1 ⊗B] f 7→ (e⊗ 1) f π0.

By a result of Cuntz [3, Cor. 1.7(b)] this map is an isomorphism for n ≥ 1.This allows us to identify from now on

KK (A,B) = [qA,K ⊗B] with limn

[qnA,Kn ⊗B].

4.3. The composition product of quasi-homomorphisms. The advan-tage of working with the direct limit is that there is a composition product

(4.4) [qmA,Km ⊗B]× [qnB,Kn ⊗ C]−→ [qm+nA,Km+n ⊗ C],

compatible with the stabilization homomorphism (4.2) which induces the de-sired composition product (1.3) on KK -groups. This composition product isdefined by sending a pair of maps f : qmA → Km ⊗ B, g : qnB → Kn ⊗ C tothe composition

qm+nA = qm(qnA)qmf−→ qm(Kn⊗B)

χmn−→ Kn⊗qmB 1⊗g−→ Kn⊗Km⊗C = Km+nC.

To describe the ∗-homomorphism χmn, it is convenient to describe the it-erated Cuntz algebra qnA directly in terms of A rather than just giving aniterative construction.

4.5. The iterated Cuntz algebra. We note that qA is a subalgebra of A∗A,hence q2A is a subalgebra of qA ∗ qA ⊂ A ∗A ∗A ∗A, e.t.c. In general, qmA isan ideal in the free product of 2m copies of A. These copies are convenientlyparameterized by the 2m subsets K of the set M = 1, . . . ,m (including K =∅ and K = M). It turns out that the obvious action of the symmetric groupΣm on the free product ∗

K⊂MAK leaves the ideal qmA ⊂ ∗

K⊂MAK invariant

thus inducing a Σm-action on qmA; this action will play a central role inour construction of the symmetric spectrum KK(A,B). To keep track of thisaction, it will be convenient to slightly generalize the iterated Cuntz algebraqmA by constructing for any finite set M a C∗-algebra qMA such that

• qMA is isomorphic to qmA if M has cardinality m and• M 7→ qMA is a functor Mop → C∗ from the opposite of the categoryM of finite sets and injective maps to the category of C∗-algebras.

We will define the C∗-algebra qMA as an ideal of the C∗-algebra

QMAdef= ∗

K⊂MAK



which is the free product of copies of A parameterized by the subsets K ⊂M(we use the superscript K in AK to keep track of the various copies of A).This C∗-algebra is generated by elements aK , where a is an element of A andK a subset of M . We define qMA ⊂ QmA to be the ideal generated by theelements

(4.6) qM (a)def=∑

K⊂M

(−1)#(K)aK , a ∈ A

where the sum is taken over all subsets of M (including M and the empty set),and #(K) is the cardinality of K.

If j : N →M is a morphism inM the corresponding map qj : qMA→ qNAis induced by the map

(4.7) Qj : QMA→ QNA aK 7−→ aj(K)

Remark 4.8. It can be shown that the ideal qMA ⊂ QMA is the intersection∩e∈M ker ρe, where ρe is the ∗-homomorphism

ρe : QMA→ QM\eA aK 7→ aK∩(M\e).

4.9. The ∗-homomorphisms ∆MN and χmn. If C, D are C∗-algebras andC ⊗ D is their spatial tensor product (cp. [11, Appendix T]) and M , N aredisjoint finite sets, we define a C∗-homomorphism

(4.10) ∆MN : QM∪N(C⊗D)→ QMC⊗QND (c⊗d)K 7→ cK∩M ⊗dK∩N .We check that ∆MN maps qM∪N (C ⊗D) ⊂ QM∪N (C ⊗D) to qMC ⊗ qND ⊂QMC ⊗QND:

∆MN (qM∪N (c⊗ d)) = ∆MN (∑

K⊂M∪N

(−1)#(K)(c⊗ d)K)

(4.11)

=∑

K⊂M∪N

(−1)#(K)cK∩M ⊗ dK∩N(4.12)

=∑

K′⊂M

∑

K′′⊂N

(−1)#(K′)+#(K′′)cK′ ⊗ cK′′

(4.13)

=

( ∑

K′⊂M

(−1)#(K′)cK′

)⊗( ∑

K′′⊂N

(−1)#(K′′)cK′′

)(4.14)

= qM (c)⊗ qN (d)(4.15)

The ∗-homomorphism

(4.16) qm(Kn ⊗B)χmn−→ Kn ⊗ qmB

is obtained by specializing ∆MN to M = ∅, N = 1, . . . ,m, and C = Kn,D = B.



5. KK as symmetric monoidal category

Let A1, A2, B1, B2 be C∗-algebras and let A1⊗A2, B1⊗B2 be their spatial(also called minimal) tensor product (cp. [11, Appendix T 5]). The tensorproduct of ∗-homomorphisms gives an associative product

(5.1) C∗(A1, B1)× C∗(A2, B2)⊗→ C∗(A1 ⊗A2, B1 ⊗B2) (f, g) 7→ f ⊗ g.

In this section we will extend this tensor product from ∗-homomorphisms to(homotopy classes of) quasi-homomorphisms to obtain an associative product

(5.2) KK (A1, B1)×KK (A2, B2)⊗→ KK (A1 ⊗A2, B1 ⊗B2).

In the language of category theory, the tensor product of C∗-algebras and∗-homomorphisms gives the category C∗ an extra structure, namely that of asymmetric monoidal category. The axioms for a symmetric monoidal category(which we will recall below) basically encode all the compatibility conditionsone might wish to impose between the composition of morphisms and theirtensor product. Similarly, saying that the tensor product (5.2) makes thecategory KK a symmetric monoidal category expresses concisely all the com-patibility conditions between the composition product in KK -theory and thetensor product in KK -theory. The compatibility between the products in C∗

and the product in KK is expressed by saying that

C : C∗ → KK

is a functor of symmetric monoidal categories.For the convenience of the reader we recall the definition of a symmetric

monoidal category, since this will be a central notion used in the followingsections. As mentioned above, this basically axiomatizes the compatibilityconditions between composition and tensor product of morphisms. The tech-nical complications come from the fact that for objects A, B, C in such acategory the object A⊗B is not equal to, but just isomorphic to B ⊗A (andsimilarly for (A ⊗ B) ⊗ C and A ⊗ (B ⊗ C)) and these isomorphisms mustcarefully be kept track of.

Definition 5.3. [2, 6.1.1] A monoidal category C consists of

(5.3.1) a category C;(5.3.2) a bifunctor m : C ×C → C, (A,B) 7→ A⊗B, called the tensor product;(5.3.3) an object I ∈ C, called the unit;(5.3.4) for every triple A,B,C of objects an associativity isomorphism

aABC : (A⊗B)⊗ C −→ A⊗ (B ⊗ C);

(5.3.5) for every object A a left unit isomorphism lA : I ⊗A −→ A;(5.3.6) for every object A a right unit isomorphism rA : A⊗ I −→ A.

The structure isomorphisms must depend naturally on the objects involved.Moreover it is required that the following two diagrams commute for objectsA,B,C,D



(5.3.A) associativity coherence

((A⊗B)⊗ C)⊗DaABC⊗1

aA⊗B,C,D// (A⊗B)⊗ (C ⊗D)

aA,B,C⊗D

(A⊗ (B ⊗ C))⊗DaA,B⊗C,D

A⊗ ((B ⊗ C)⊗D)1⊗aB,C,D

// A⊗ (B ⊗ (C ⊗D))

(5.3.U) unit coherence

(A⊗ I)⊗B aAIB //

rA⊗1&&MMMMMMMMMM

A⊗ (I ⊗B)

1⊗lBxxqqqqqqqqqq

A⊗BDefinition 5.4. [2, 6.1.2] A monoidal category is called symmetric if in addi-tion for every pair A,B there is a symmetry isomorphism

sAB : A⊗B → B ⊗Awhich depends naturally in A and B. The symmetry isomorphisms must becompatible with the other structure isomorphisms in the sense that the follow-ing two diagrams commute for any choice of objects A,B,C


(A⊗B)⊗ C sAB⊗1//

aABC

(B ⊗A)⊗ CaBAC

A⊗ (B ⊗ C)

sA,B⊗C

B ⊗ (A⊗ C)

1⊗sAC

(B ⊗ C)⊗A aBCA // B ⊗ (C ⊗A)


A⊗ I sAI //

rA""EEEEEEEE

I ⊗A

lA||yyyyyyyy

A

and in addition they must satisfy the symmetry axiom, i.e. for any two objectsA,B the composite sBA sAB is the identity in C(A⊗B,A⊗B).

Example 5.5. The category Ab of abelian groups with the usual tensor prod-uct is a symmetric monoidal category (the unit I is the group Z and all thestructure isomorphisms are the obvious ones).



Example 5.6. The category C∗ of C∗-algebras over F = C or F = R withthe spatial tensor product is a symmetric monoidal category (the unit I is theC∗-algebra F and again all the structure isomorphisms are the obvious ones).

Example 5.7. The category of pointed compactly generated weak Hausdorffspaces Top∗ with the tensor product being the smash product and the well-known structure isomorphisms define a symmetric monoidal category. Whenworking with the symmetric monoidal category Top∗ various constructions(e.g. like taking suitable mapping spaces) yield weak Hausdorff spaces whichare not compactly generated. However there is an idempotent functor whichproduces out of a given topology a coarsest topology which contains the givenone and is compactly generated. We therefore will tactically assume that allpointed (weak Hausdorff) spaces we are considering are first hit by this functor,so we can regard them as objects in Top∗.

5.8. Tensor product of quasi-homomorphisms. Now we define a tensorproduct

(5.9) [qmA1,Km⊗B1]×[qnA2,Kn⊗B2]⊗−→ [qm+n(A1⊗A2),Km+n⊗B1⊗B2]

which sends a pair (f1, f2) of ∗-homomorphisms to the composition

qm+n(A1 ⊗A2)∆mn

−→ qmA1 ⊗ qnA2

f1⊗f2−→ (Km ⊗B1)⊗ (Kn ⊗B2)∇mn−→ Km+n ⊗B1 ⊗B2.

Here ∇mn is the obvious ∗-isomorphism involving shuffling of the factors andthe canonical isomorphism Km ⊗ Kn ∼= Km+n; the ∗-homomorphism ∆mn

is equal to ∆MN for M = 1, . . . ,m, N = m + 1, . . . ,m + n using theidentifications qmA = qMA, qnA = qNA, qm+nA = qM∪NA.

We observe that this product in fact agrees with the tensor product of∗-homomorphisms for m = n = 0. Moreover, this product is compatible withthe stabilization homomorphism (4.2) and hence induces the desired tensorproduct (5.2) on KK -groups. (To see the latter one can argue as in [6, 4.7]).

6. Enriched categories

In the previous two sections we have investigated the category KK . Bydefinition of a category for any two objects A,B one has a corresponding setof morphisms from A to B. We have seen that the morphism sets KK (A,B)for C∗-algebras A and B are abelian groups, and that the composition is abilinear map. In categorical language one would say that KK is an preaddi-tive category. Alternatively one could say that the category KK is enrichedover the symmetric monoidal category of abelian groups. Conceptually an en-richment of a category D over a symmetric monoidal category C is given byidentifying the morphism sets D(A,B) with an object in C in such a way thatthe symmetric monoidal product of the category C can be used to describe thecomposition. Below we will see that KK also can be given an enrichment overthe category of pointed spaces.



Before we define the precise definition of an enrichment we need to introducethe notion of an enriched category.

Definition 6.1. ([2, 6.2.1]) Let C be a monoidal category as defined in Defi-nition 5.3. A C-category D (or an enriched category) consists of the followingdata:

(6.1.1) a class of objects |D|;(6.1.2) for every pair of objects A,B ∈ |D| an object D(A,B) ∈ |C|.(6.1.3) for every triple of objects A,B,C ∈ |D| a composition morphism

cABC : D(A,B)⊗D(B,C) −→ D(A,C);

(6.1.4) for every object A ∈ |D| a unit morphism uA : I → D(A,A),

and the structure maps are required to yield the following commutative dia-grams for objects A,B,C,D ∈ |D|(6.1.A) associativity coherence

(D(A,B) ⊗D(B,C)) ⊗D(C,D)cABC⊗1

//

aD(A,B)D(B,C)D(C,D)

D(A,C) ⊗D(C,D)

cACD

D(A,B)⊗ (D(B,C) ⊗D(C,D))

1⊗cBCD

D(A,B)⊗D(B,D)cABD // D(A,D)


I ⊗D(A,B)lD(A,B)

//

uA⊗1

D(A,B)

1

D(A,B)⊗ IrD(A,B)

oo

1⊗uB

D(A,A)⊗ D(A,B)cAAB // D(A,B) D(A,B) ⊗D(B,B)

cABBoo

Example 6.2. (cp. [2, 6.2.9]) Let D be a C-category as in the previous defini-tion. Assume further that the monoidal product of C is symmetric. Then theC-category D ⊗D is defined by the following data:

(6.2.1) |D ⊗ D|=|D| × |D|;(6.2.2) for a pair of objects (A,A′), (B,B′) ∈ |D ⊗ D| the morphism object is

given by (D ⊗D)((A,A′), (B,B′)) = D(A,B) ⊗D(A′, B′).(6.2.3) for every triple of objects (A,A′), (B,B′), (C,C′) ∈ |D ⊗ D| the com-

position morphism is given by the obvious map

(D(A,B) ⊗D(A′, B′))⊗ (D(B,C) ⊗D(B′, C′)) −→ D(A,C) ⊗D(A′, C′);

(6.2.4) for every object (A,A′) ∈ |D ⊗ D| the unit morphism is given by

u(A,A′) : Ir−1I−→ I ⊗ I uA⊗uA′−→ D(A,A) ⊗D(A′, A′),



Definition 6.3. ([2, 6.2.3]) A functor F : D → E between C-categories consistsof the following:

(6.3.1) for every object A ∈ |D| an object FA ∈ |E| and(6.3.2) for every pair of objects A,A′ ∈ |D| a morphism

FA,A′ : D(A,A′)→ E(FA,FA′)such that or all objects A,A′, A′′ ∈ |D| the following diagrams commute

(6.3.N) naturality condition

D(A,A′)⊗D(A′, A′′)

FAA′⊗FA′,A′′

cAA′A′′// D(A,A′′)

FA,A′′

E(FA,FA′)⊗ E(FA′, FA′′) cFA,FA′,FA′′

// E(FA,FA′′)

(6.3.U) unit condition

IuA

//

uFA$$IIIIIIIIII D(A,A)

FAA

E(FA,FA)

Note that any category naturally has the structure of a Set-category. On theother hand, an enriched category is not a category as the morphism objects ingeneral cannot be interpreted as sets. This however can be done after choosinga lax symmetric monoidal functor F : C → Set.

Definition 6.4. (cp. [2, 6.4.3]) Let F : C → C′ be a lax monoidal functorbetween monoidal categories, i.e. F : C → C′ is a functor which comes equippedwith a natural transformation of bifunctors

F (V )⊗ F (W ) −→ F (V ⊗W ), V,W ∈ |C|and a unit morphism IC′ → F (IC) such that all coherence diagrams relatingassociativity and unit isomorphisms of C and C′ are commutative. Given aC-category D the lax monoidal functor F can be used to define a C′-categoryF∗D. The latter is given by the following data:

(6.4.1) the class of objects is |F∗D| = |D|;(6.4.2) for a pair of objects A,B the morphism object is

F∗D(A,B) = F (D(A,B)).

(6.4.3) for a triple of objects A,B,C the composition morphism is given by

F (D(A,B)) ⊗ F (D(B,C)) −→ F (D(A,B) ⊗D(B,C))F (cABC)−→ F (D(A,C));

(6.4.4) for an object A the unit morphism uA : IC′ → F (IC)F (uA)→ F (D(A,A)).



An enrichment of a category D′ over a symmetric monoidal category C is atriple consisting of a C-category D, a lax symmetric monoidal functor F : C →Set and an isomorphism F∗D ∼= D′. In various examples to be discussed thecategory C will be equipped with a forgetful functor to Set. In these casesit should be understood that the corresponding lax monoidal functor we areusing is the forgetful functor, unless explicitly stated otherwise.

6.5. Enrichments of the category KK . As already mentioned the cat-egory KK has an enrichment over the monoidal category of abelian groups.It can also be enriched over the monoidal category Top∗ of pointed spaces(introduced in 5.7) using the Cuntz picture. This goes as follows. First weequip the category of C∗-algebras with an enrichment over the category ofpointed topological spaces. Let us define Hom(A,B) to be the pointed set of∗-homomorphisms from a C∗-algebra A to a C∗-algebra B with the compactopen topology, the basepoint being the zero homomorphism. The compositionof ∗-homomorphisms then yields a continuous map

cABC : Hom(A,B) ∧Hom(B,C) −→ Hom(A,C)

for C∗-algebras A,B,C. Finally the unit morphisms uA : S0 → Hom(A,A) aregiven by requiring the image of uA to be 0, idA ⊂ Hom(A,A). One easilychecks that these data define an enrichment of the category of C∗-algebrasover the category Top∗. Obviously, if we apply the forgetful functor to themorphism spaces we get back the ordinary category of C∗-algebras and ∗-homomorphisms.

The enrichment of the category of C∗-algebras and ∗-homomorphisms overthe category Top∗ induces a corresponding enrichment of the category KK .The morphism spaces of the corresponding Top∗-category KK top are given by

KK top(A,B) = colimm

Hom(qmA,K⊗m ⊗B).

where the structure maps for the colimit are defined as in (4.2). The com-position is defined as described in 4.3. The Top∗-category KK top then is anenrichment of the category KK by means of the functor F = π0 : Top∗ →Set,X 7→ π0(X).

Next recall that the category KK can be regarded as a symmetric monoidalcategory by means of the external product

KK (A,B)⊗KK (A′, B′)→ KK (A⊗A′, B ⊗B′),for C∗-algebras A,A′, B,B′. The fact that these maps are bilinear means thatthe product is compatible with the enrichment over KK over the categoryof abelian groups. In category language this can be phrased by saying thatthe external product gives the enriched category KK the structure of an en-riched symmetric monoidal category (cp. 6.9). Conceptually the definition ofan enriched symmetric monoidal category is analogous to the definition of asymmetric monoidal category. Essentially one just needs to replace the role ofthe category by an enriched category. However, to make this explicit one needs



to say what a morphism and what an isomorphism in an enriched category is.Furthermore one needs to say what it means that data depend naturally onthe objects.

Definition 6.6. Let C be a monoidal category. Let D be an enriched categoryas defined in Definition 6.1, and let A,B,C be objects of D. A morphism fromA to B we define to be a map f : I → D(A,B) in C; in symbols we writef : A → B. Given a morphism f : A → B and a morphism g : B → C thecomposite g f : A→ C is given by

Ir−1I // I ⊗ I f⊗g

// D(A,B)⊗D(B,C)cABC // D(A,C).

In particular this convention allows to consider commutative diagrams of mor-phisms in enriched categories.1 Moreover we can define an isomorphism in Dto be a morphism f : A → B for which there is a morphism g : B → A suchthat g f = uA and f g = uB.

Definition 6.7. Given an enriched category D in the sense of Definition 6.1,two functors of enriched categories F,G : D → D and a family of morphismsfA : F (A)→ G(A), then we say that the family is natural in A if for any twoobjects A,B ∈ |D| we have a commutative diagram

D(A,B)⊗ I F⊗fB// D(F (A), F (B)) ⊗D(F (B), G(B))

cF (A),F (B),B

D(A,B)

r−1D(A,B)

88ppppppppppp

l−1D(A,B) &&NNNNNNNNNNN

D(F (A), G(B))

I ⊗D(A,B)fA⊗G

// D(F (A), G(A)) ⊗D(G(A), G(B))

cF (A),G(A),G(B)

OO

It then follows that the diagrams

F (A)

F (f)

fA// G(A)

G(f)

F (B)fB

// G(B)

commute for all morphisms f : A → B between objects A and B of D. Theconverse is not true in general. However it is true if I is a generator2 of thecategory D.

1The composition of morphisms in the sense of Definition 6.6 is associative, which isa consequence of the identity lI = rI . The latter follows from the axioms of a monoidalcategory, see [9, Theorem 3].

2A generator of a category D is an object G such that for any pair of morphism f, g :A → B in the category D the following holds: g = f if and only if g h = f h for allmorphism h ∈ D(G, A).



Example 6.8. Let Set denote the symmetric monoidal category of sets withthe tensor product being the cartesian product. Any category then is a Set-category in a natural way. The morphisms of a category precisely correspondto morphisms (in the sense of Definition 6.6) of the corresponding Set-category.

Definition 6.9. Let C be a monoidal category with a symmetric monoidalproduct. A monoidal C-category consists of the following data:

(6.9.1) a C-category D;(6.9.2) a bifunctor of enriched categories m : D ⊗D → D, (A,B) 7→ A⊗B,(6.9.3) an object U ∈ |D|, called the unit;(6.9.4) for every triple A,B,C of objects an associativity isomorphism

aABC : I → D((A ⊗B)⊗ C,A⊗ (B ⊗ C));

(6.9.5) for every object A a left unit isomorphism lA : I → D(U ⊗A,A);(6.9.6) for every object A a right unit isomorphism rA : I → D(A⊗ U,A).

The structure isomorphisms must naturally depend on the objects. Moreoverit is required that the structure isomorphism yield commutative associativitycoherence diagrams (5.3.A) as well as commutative unit coherence diagrams(5.3.U). Furthermore for all objects A,B,C,A′, B′, C′ ∈ |D| the following dia-grams (where the unlabeled maps are induced by the structure maps) have tobe commutative

(6.9.A) associativity condition

(D(A,A′)⊗ D(B,B′))⊗D(C,C′)m⊗1

//

aD(A,A′),D(B,B′),D(C,C′)

D(A⊗B,A′ ⊗B′)⊗D(C,C′)

m

D(A,A′)⊗ (D(B,B′)⊗D(C,C′))

1⊗m

D((A⊗B)⊗ C, (A′ ⊗B′)⊗ C′)

D(A,A′)⊗D(B ⊗ C,B′ ⊗B′) c // D(A ⊗ (B ⊗ C), A′ ⊗ (B′ ⊗ C′))(6.9.U) unit condition

I ⊗D(A,B)lD(A,B)

//

uA⊗1

D(A,B)

1

D(A,B) ⊗ IrD(A,B)oo

1⊗uB

D(U,U)⊗D(A,B)

m

D(A,B) ⊗D(U,U)

m

D(U ⊗A,U ⊗B) // D(A,B) D(A⊗ U,B ⊗ U)oo

Definition 6.10. Let C be a monoidal category with a symmetric monoidalproduct. A monoidal C-category D is called symmetric if in addition for everypair of objects A,B ∈ D there is a symmetry isomorphism

sAB : I → D(A ⊗B,B ⊗A)



which depends naturally on A and B. The symmetry isomorphism must becompatible with the other structure isomorphisms in the sense that all the as-sociativity coherence diagrams (5.4.A) and all unit coherence diagrams (5.4.U)commute. Moreover they must satisfy the symmetry axiom, i.e. for any two ob-jects A,B the composite sBAsAB is the unit morphism I → D(A⊗B,A⊗B).Furthermore for all A,A′, B,B′ ∈ |D| the following diagram (where the verticalmorphism is induced by the symmetry isomorphism) has to be commutative

(6.10.S) symmetry condition

D(A,A′)⊗D(B,B′)

sD(A,A′),D(B,B′)

m // D(A⊗B,A′ ⊗B′)

D(B,B′)⊗D(A,A′)m // C(B ⊗A,B′ ⊗A′)

Let F : C → C′ be a lax (symmetric) monoidal functor of (symmetric)monoidal categories and let D be a (symmetric) monoidal C-category. Wethen can extend Definition 6.4 in a straightforward manner in order to obtaina (symmetric) monoidal C′-category F∗D. Accordingly we can define the no-tion of an enrichment of a (symmetric) monoidal category over a (symmetric)monoidal category C.6.11. Monoidal enrichments of the category KK . As already mentioned,from the bilinearity of the exterior Kasparov product (5.9) it follows thatthe symmetric monoidal category KK has an enrichment over the symmetricmonoidal category of abelian groups. One may check that the maps

m : KK top(A1, B1)⊗KK top(A2, B2) −→ KK top(A1 ⊗A2, B1 ⊗B2)

given by the maps introduced in 5.8 turn KK top into an enrichment of themonoidal category KK over the symmetric monoidal category Top∗. Howeverthe enriched monoidal Top∗-category KK top is not symmetric. To see this itsuffices to look at the following diagrams

q2(A1 ⊗A2) //

q2(sA1,A2)

qA1 ⊗ qA2

sqA1 ,qA2

q2(A2 ⊗A1) // qA2 ⊗ qA1

(K ⊗A1)⊗ (K ⊗A2) //

sK⊗A1,K⊗A2

K⊗2 ⊗A1 ⊗A2

id⊗sA1,A2

(K ⊗A2)⊗ (K ⊗A1) // K⊗2 ⊗A2 ⊗A1

They do not commute, and from this one easily sees that the exterior producton KK top cannot be symmetric. A similar lack of symmetry also shows upquite prominently in stable homotopy theory. In stable homotopy theory thelack of symmetry was resolved by introducing the category of symmetric spacesand spectra.

7. Symmetric spaces and spectra

Let ℘ denote the small category whose set of objects is the set of finitesubsets of the natural numbers and whose sets of morphisms ℘(M,N) for two



subsets M,N ⊂ N consists of the set maps from M to N , i.e. ℘(M,N) =Set(M,N). Let I denote the subcategory consisting of the isomorphisms. Afunctor from I into a categoryD is called a symmetric sequence inD. Occasion-ally we also call a symmetric sequence in a category D just a symmetric object.Our major interest will be in the category of symmetric spaces and spectra,defined below.

Definition 7.1. LetD be a monoidal category with an initial object ∗. Assumefurther that D has finite coproducts and that the tensor product preserves thefinite coproducts (up to natural coherence). The category DI of functors fromI to D then carries the structure of a monoidal category. The correspondingdata are given by

(7.1.1) the underlying category is DI , the objects in DI are called symmetricsequences in D;

(7.1.2) the bifunctor ⊗ : DI × DI → DI for symmetric sequences X,Y ∈ DIis given by

(X ⊗ Y )(J) =∐

M∪N=JM∩N=∅

X(M)⊗ Y (N), J ∈ |I|;

(7.1.3) the unit E, given by E(∅) = I and E(K) = ∗, the initial object, forK 6= ∅;

(7.1.4) the associativity isomorphism aXY Z : (X ⊗ Y ) ⊗ Z −→ X ⊗ (Y ⊗ Z)is given by the composite

((X ⊗ Y )⊗ Z)(J)= //

∐L∪K=JK∩L=∅

∐M∪N=LM∩N=∅

X(M)⊗ Y (N)

⊗ Z(K)

∼=

∐M∪N∪K=J

M∩N=N∩K=K∩M=∅

(X(M)⊗ Y (N))⊗ Z(K)

∐M∪N∪K=J

M∩N=N∩K=K∩M=∅

X(M)⊗ (Y (N)⊗ Z(K))

(X ⊗ (Y ⊗ Z))(J)= //

∐M∪L=JM∩L=∅

X(M)⊗

∐N∪K=LN∩K=∅

Y (N)⊗ Z(K)

with the left and the right unit isomorphism induced by the left and the rightunit isomorphism of the monoidal categoryD. The categoryDI has a canonical



initial object: it is given by the constant functor which sends every object inI to the initial object ∗.

If D is a symmetric monoidal category the category DI also can be given asymmetric monoidal structure; the symmetry isomorphism sXY for symmetricsequences X,Y in D is the map X ⊗ Y → Y ⊗ X whose restriction to thefactors X(M)⊗ Y (N) ⊂ (X ⊗ Y )(K) is given by the composites

X(M)⊗ Y (N)sX(M)Y (N)

// Y (N)⊗X(M) ⊂ (Y ⊗X)(K).

If D is an enriched symmetric monoidal category enriched over a symmetricmonoidal category C which contains all small limits then DI also is enrichedover C: the morphism object DI(X,Y ) for two symmetric objects X,Y ∈ |DI |is

DI(X,Y ) = limM∈I

D(X(M), Y (M))

In any symmetric monoidal category there is the notion of a monoid andthe notion of modules over a monoid.

Definition 7.2. A monoid in a monoidal category D is an object R ∈ |D|together with a multiplication map µ : R ⊗R → R and a unit map η : I → Rfor which the following diagrams are commutative


(R⊗R)⊗Rµ⊗1

aRRR // R⊗ (R⊗R)

µ

R⊗R µ// R R⊗Rµ

oo


R⊗ I η⊗1//

lR$$JJJJJJJJJJ

R⊗Rµ

R⊗ I1⊗ηoo

rRzztttttttttt

R

If D is symmetric monoidal category a monoid is called commutative if µ =sRRµ.

Definition 7.3. A left module over a monoid R in a monoidal category D isan object M ∈ D together with a map ν : R⊗M →M such that the followingdiagram is commutative

(R⊗R)⊗M aRRM //

µ⊗1

R⊗ (R⊗M)1⊗ν

// R⊗Mν

I ⊗Mη⊗Moo

lMyytttttttttt

R⊗M ν // M

Similarly one can define a right module over R. If D is a symmetric monoidalcategory and R is a commutative monoid in D any left module over R can be



given the structure of a right module over R by defining the right action to bethe composite νsMR : M ⊗R→M .

Lemma 7.4 (Hovey-Shipley-Smith, Lemma 2.2.2 & Theorem 2.2.10). Let D bea symmetric monoidal category that is cocomplete and let R be a commutativemonoid in D such that the functor R ⊗ : D → D preserves coequalizers.Then there is a symmetric monoidal product ⊗R on the category of left R-modules with R as unit. For two left R-modules X,Y the product is given bythe coequalizer described by the diagram

X ⊗R⊗ YµsXR⊗1−−−−−→−−−−−→

1⊗µ

X ⊗ Y −→ X ⊗R Y.

7.5. Symmetric spectra. From now on we will specify to the special case ofour major interest which is the case where D is the category Top∗ of pointedspaces.

Definition 7.6. Let S denote the monoid in pointed symmetric spaces whichis given by the functor M 7→ (S1)∧M and the obvious structure maps. Asymmetric spectrum is a left S-module.

Definition 7.7. A commutative symmetric ring spectrum is a map of commu-tative monoids S → R. A (left) R-module canonically inherits the structure ofa symmetric spectrum by means of the monoid map. By the previous lemmathe category of (left) R-module spectra has a symmetric monoidal product ∧Rwith unit R.

The definition of a symmetric spectrum given above is not the standard def-inition (cp. [7]). However the category of symmetric spectra as we defined it isequivalent to category of symmetric spectra defined via the standard definition.In both approaches one uses a diagram category of “symmetric sequences”; thedifference between the standard and our approach is that we use a bigger butequivalent diagram category. More precisely, in the standard set-up one usesthe full subcategory Σ ⊂ I with objects the sets n = 1, 2, . . . , n, n ∈ N.

Definition 7.8. Let E be a symmetric spectrum. For a natural number n ∈ Nlet n also denote the subset n = 1, 2, 3, . . . , n ⊂ N. The 0-th homotopy groupof E is defined by

(7.9) π0(E) = colimn

πn(E(n)),

where the structure maps are induced by the natural inclusions n ⊂ n+ 1.

8. Enrichments over symmetric spaces

In this section we will define two categories which are enriched over thecategory of symmetric spaces, KK and KK (Theorem 8.7 and Theorem 8.13).In Section 10 we will see that KK has an enrichment over the category ofsymmetric spectra.



8.1. The enriched category KK. In 4.5 we have seen that the iteratedCuntz construction q•A for a C∗-algebra A defines a contravariant functorfrom the category I of finite subsets of N and isomorphisms to the category ofC∗-algebras. On the other hand given a C∗-algebra B we have the symmetricC∗-algebra

K•B : M 7−→ KM ⊗B.These functors define a symmetric space

(8.2) KK(A,B) : M −→ KKM (A,B) = Hom(qMA,KMB),

where we use the compact-open topology to topologize the sets of ∗-homomor-phisms. We thus have a bivariant functor from the category of C∗-algebras tothe category of symmetric spaces. We shall see that they are the morphismobjects of an enriched category. We need to define the composition morphisms.

Let M,N ∈ ℘ be subsets with M ∩ N = ∅. Recall from Section 4.9 thedefinition of the map ∆MN . For a C∗-algebra B we used ∆∅M to define

(8.3) χMN : qM (KN ⊗B)∆∅M

−→ KN ⊗ qMB.For f ∈ Hom(qMA,KM ⊗B) and g ∈ Hom(qNB,KN ⊗C) define cABC(f, g) ∈Hom(qM∪NA,KM∪NC) as the composition

qM∪NA∼=−→ qNqMA

qNf−→ qN (KM ⊗B)χMN

−→ KM ⊗ qNB id⊗g−→ K⊗M∪N ⊗C.This defines a map

cABC : Hom(qMA,KMB) ∧Hom(qNB,KNC)→ Hom(qM∪NA,KM∪NC),

and varying the subsets M,N yields a corresponding map

(8.4) cABC : KK(A,B)⊗KK(B,C) −→ KK(A,C).

We claim that these maps define an enriched category KK. To check thisstatement involves quite a bit of combinatorics and we will derive it from ageneral recipe.

8.5. Composition data. Let C be a symmetric monoidal category with aninitial object ∗. Assume further that C has finite coproducts and that thetensor product preserves the finite coproducts (up to natural coherence). LetD be a C-category which is an enrichment of an ordinary category by means of afaithful forgetful functor to Set. This means that we can regard the morphismobjects of D as sets.3 For a functor F into the category DI let FM denote theevaluation of the functor on an object M ∈ |I|.

Let F,G : D → DI be two functors of enriched categories. Assume that thefunctors F,G : D → DI are augmented in the sense that F∅ = id = G∅. Aset of composition data for the pair (F,G) consists of natural transformationsof functors D → DMN : FM∪N → FMFN ; ιMN : GMGN → GM∪N ; χMN : FMGN → GNFM

3This is a technical assumption which leads to a simplification of the statement and proofof the following proposition. An analogous statement also holds without this assumption.



for any pair of finite subsets M,N ⊂ N with M ∩ N = ∅. These functorsmust naturally depend on M and N . Moreover for any triple of pairwisedisjoint finite subsets L,M,N ⊂ N the natural transformations must yieldcommutative diagrams

FL∪M∪N(L∪M)N

//

L(M∪N)

FL∪MFN

LM1

GLGMGN1ιMN

//

ιLM1

GLGM∪N

ιL(M∪N)

FLF (M∪N)1MN

// FLFMFN GL∪MGNι(L∪M)N

// GL∪M∪N

FL∪MGNLM1

//

χ(L∪M)N

FLFMGN

1χMN

FLGMGN1ιMN

//

χML1

FLGM∪N

χL(M∪N)

FLGNFM

χLN1

GMFLGN

1χLN

GNFL∪M1L∪M

// GNFLFM GMGNFLιMN1

// GM∪NFL

In addition the maps MN , χMN and ιMN must be the identity if either of Mor N is the empty set.

Proposition 8.6. Let us be given composition data as defined in 8.5. Thenwe can define a CI-category D as follows

(8.6.1) |D| = |D|;(8.6.2) for each pair of objects A,B ∈ |D| the object D(A,B) ∈ |CI | is given

by

D(A,B) : M 7→ DM (A,B) = D(FM (A), GM (B)),M ∈ |I|;(8.6.3) for every triple of objects A,B,C ∈ |D| the composition morphism

cABC : D(A,B) ⊗ D(B,C) → D(A,C) is the unique morphism deter-mined by the maps

D(FN (A), GN (B)) ⊗D(FM (B), GM (C)) −→ D(FM∪N (A), GM∪N (C)),

for disjoint finite subsets M,N ⊂ N, which sends an element f ⊗ g tothe map

cABC(f ⊗ g) : FM∪N (A)MN→ FMFN (A)

FM (f)→ FMGN (B)

χMN−→ GNFM (B)GN (g)→ GNGM (C)

ιMN→ GM∪N (C);

(8.6.4) for an object A ∈ |D| the unit morphism uDA : E → D(A,A) is deter-

mined by (uDA)(∅) = uDA : I → D(A,A) = D∅(A,A).

The category D is an enrichment over the category D by means of the forgetfulfunctor which associates to a symmetric D-object X the D-object X(∅).



Proof. Let L,M,N be pairwise disjoint finite subsets of N, and let us be givenf ∈ D(FN (A), GN (B)), g ∈ D(FM (B), GM (C)) and h ∈ D(FL(C), GL(D)).The following diagram (with the obvious maps) commutes and thereby showsthat the associativity coherence conditions (6.1.A) hold in D.

FL∪M∪N (A)= //

FL∪M∪N (A)

FLFM∪N (A)

// FLFMFN (A)

FL∪MFN (A)

oo

FLFMGN (B)

FL∪MGN (B)

oo

FLGNFM (B)

// GNFLFM (B)

GNFL∪M (B)oo

FLGM∪N (C)

FLGNGM (C)oo // GNFLGM (C)

GM∪NFL(C)

GNGMFL(C)

oo

GM∪NGL(D)

GNGMGL(D) //oo GNGL∪M (D)

GL∪M∪N(D)= // GL∪M∪N (D)

The unit coherence conditions (6.1.U) for D are fulfilled if and only if thefollowing equations hold

cAAB(1A ⊗ g) = g for all g ∈ DM (A,B), M ∈ |I|, A,B ∈ |D|;cABB(f ⊗ 1B) = f for all f ∈ DM (A,B), M ∈ |I|, A,B ∈ |D|.

These equations follow immediately from the assumption that the functors F •

and G• are augmented and the last property mentioned in 8.5.

Theorem 8.7. The following data define an enriched category KK:

(8.7.1) |KK| is the class of C∗-algebras;(8.7.2) for a pair of C∗-algebras A,B the morphism object is the symmetric

space KK(A,B) defined in (8.2)(8.7.3) for every triple of C∗-algebras A,B,C the composition morphism cABC

is given by (8.4).



(8.7.4) for all C∗-algebras A the unit morphism uA : E → KK(A,A) is the onedetermined by ((uA)(∅))(S0) = 0 ∪ 1A ⊂ Hom(A,A) = KK∅(A,A).

The enriched category KK is an enrichment of the category of C∗-algebrasover the category of symmetric spaces by means of the forgetful functor whichassociates to a symmetric space X• the space X∅.

Proof. Put F •A = q•A,G•B = K•B. For any pair (M,N) of disjoint finitesubsets of N we then have canonical isomorphisms MN : qM∪NA ∼= qMqNAand ιMN : KMKNB ∼= KM∪NB, as well as the natural transformation χMN

defined in (8.3). It now is straight forward to check the compatibility conditionsintroduced in 8.5. The assertion then follows from the previous proposition.

8.8. (Co)associative functors. The notion of composition data introducedin 8.5 can be regarded as a collection of several pieces of information which alsocan be looked at separately. For example it makes sense to consider the pairF = (F •, ) consisting of the functor F • and the natural transformation andjust require the upper right diagram in the definition to be commutative. Sucha structure we might call an coassociative functor. Dually the pair G = (G•, ι)consisting of the functor G• and the natural transformation ι subject to thecommutativity of the upper right diagram we might call an associative functor.The natural transformation χ then is a sort of intertwining operator between Fand G and the two diagrams in the lower row correspond to the compatibilityof the intertwining operator χ with the natural transformations and ι whichdefine the coassociative and the associative structure respectively. Completelyanalogously one could define composition data for two coassociative functors F1

and F2 or for two associative functors G1 and G2. Corresponding compositiondata would imply that one can define functors (F1F2)

•, (G1G2)• : D → DI

given by M 7→ FM1 FM2 (A) and M 7→ GM1 GM2 (B) which then would comeequipped with the structure of a (co-)associative functor. Furthermore, if thereare given composition data for two coassociative functors F1 and F2 with anatural transformation χMN : FM1 FN2 → FN2 FM1 as well as composition datafor F1 and G and for F2 and G for an associative functor G then this datadefine composition data for (F1F2) and G. Similarly given two associativefunctors G1 and G2 and a coassociative functor F together with correspondingcomposition data then one obtains from this composition data for the functorsF and (G1G2). This recipe sometimes simplifies checking the commutativityof the relevant diagrams for specific composition data.

We now give the definition of the second enrichment which eventually willgive the desired enrichment over the category of symmetric spectra. To mo-tivate the construction we recall the following variant of the Bott periodicitytheorem.

Theorem 8.9. (Bott periodicity) For any n ∈ N and all C∗-algebras A,Bthe exterior multiplication with the element 1C0(Rn) ∈ KK (C0(Rn), C0(Rn))



yields an isomorphism

(8.10) KK (A,B) −→ KK (C0(Rn)⊗A,C0(Rn)⊗B).

Via the canonical isomorphism Hom(A,C0(X)⊗B) ∼= map∗(X,Hom(A,B))for locally compact spaces X and C∗-algebras A and B we obtain from theBott periodicity theorem for all n ∈ N an isomorphism

πn(Hom(qn(C0(Rn)⊗A),KnB))∼= πn(Ωn Hom(qn(C0(Rn)⊗A),KnC0(Rn)⊗ B))∼= π0(Ω

n Hom(qn(C0(Rn)⊗A),KnB))∼= π0(Hom(qn(C0(Rn)⊗A),KnC0(Rn)⊗B))

Thus Hom(qn(C0(Rn) ⊗ A),KnB) qualifies as the n-th space of a spectrumwhich represents KK (A,B). In view of this observation we introduce

8.11. The enriched category KK. Let (qC)•A for a C∗-algebra A denotethe symmetric C∗-algebra with

(qC)MA = qM (C0(RM )⊗A), M ∈ |I|.For C∗-algebras A,B define the symmetric pointed space

KK(A,B) : M 7−→ KKM (A,B) = Hom((qC)MA,KMB).

For disjoint subsets M,N ⊂ N and C∗-algebras A,B,C we have the map

KKM (A,B) ∧KKN(B,C)→ KKM∪N (A,C)

which sends f ∧ g ∈ Hom((qC)MA,KMB) ∧ Hom((qC)NA,KNB) to the fol-lowing composition4 in Hom((qC)M∪NA,KM∪NB)

qM∪N (C0(RM∪N )⊗A)MN

→ qMqN (C0(RM )⊗ C0(RN )⊗A)qM (∆∅N )−→

qM (C0(RM )⊗ qN (C0(RN )⊗A))qM (id

C0(RM )⊗f)−→ qM (C0(RM )⊗KNB)

∼=−→

qM (KNC0(RM )⊗B)χMN

−→ KNqM (C0(RM )⊗B)KNg−→ KNKMC ιMN

−→ KN∪MC;

where the ∗-homomorphisms MN , χMN , ιMN are as in the proof of Theorem8.7; ∆∅N has been defined in 4.9. These maps define a map

(8.12) cABC : KK(A,B) ∧KK(B,C) −→ KK(A,C).

Theorem 8.13. The following data define an enriched category KK:

(8.13.1) |KK| is the class of C∗-algebras;(8.13.2) for a pair of C∗-algebras A,B the morphism object is KK(A,B)(8.13.3) for every triple of C∗-algebras A,B,C the composition morphism is

the map (8.12)(8.13.4) for a C∗-algebras A the unit morphism uA : E → KK(A,A) is the one

determined by (uA)∅(S0) = 0 ∪ 1A ⊂ Hom(A,A) = KK∅(A,A).

4The composition can be written down more compactly using the ∗-homomorphism∆MN ; this presentation however fits better with the strategy of the proof Theorem 8.13.



Proof. For disjoint finite subsets M,N ⊂ N define a map ˜MN by

(qC)M∪NAMN

→ qMqN (C0(RM∪N )⊗A) ∼= qMqN (C0(RM )⊗ C0(RN )⊗A)

qM (∆∅N )→ qM (C0(RM )⊗ qN (C0(RN )⊗A)) = (qC)M (qC)NA.

Furthermore define maps χMN by

(qC)MKNA = qM (C0(RM )⊗KNA) ∼=

qM (KNC0(RM )⊗A)χMN

→ KNqM (C0(RM )⊗A) = KN (qC)MA,

and let the maps ιMN = ιMN : KMKNA→ KM∪NA be the canonical isomor-phisms. These maps define composition data for the functors (qC)• and K• inthe sense of 8.5. The commutativity of the relevant diagrams can be checkeddirectly, or one can use the recipe that we introduced in 8.8. Proposition 8.6then yields an enrichment of the category of C∗-algebras. It is straightforwardto check that the composition morphism that one obtains from Proposition 8.6coincides with the definition of cABC as given by (8.12).

9. Symmetric monoidal enrichments over symmetric spaces

Next we want to show that the exterior Kasparov product gives KK andKK (introduced in the previous section) the structure of an enriched symmetricmonoidal category. As the category KK is the category of preferred interestwe only give the details for this case. The main result we are after is Theorem9.5.

9.1. The bifunctor ⊗ : KK ∧ KK → KK. To put a symmetric monoidalstructure on KK requires the definition of a bifunctor ⊗ : KK ∧ KK → KK.Recall from Definition 6.3 that a functor between enriched categories is givenby two pieces of data. In the situation at hand these are given by the following

(9.1.1) To a pair of C∗-algebras (A,B) we certainly associate its (spatial)tensor product A⊗B. After all we want to have an enrichment of thesymmetric monoidal category of C∗-algebras;

(9.1.2) For two pairs of C∗-algebras (A,B), (A′, B′) we define the correspond-ing morphism KK(A,B)∧KK(A′, B′)→ KK(A⊗A′, B⊗B′) throughthe individual maps

hom(qM (C0(Rn)⊗A),KMB) ∧ hom(qN (C0(RN )⊗A′,KNB′)−→ hom(qM∪N (C0(RM∪N )⊗A),KM∪NB ⊗B′)



given by sending a pair of ∗-homomorphisms f ∧ g to the followingcomposition

qM∪N (C0(RM∪N )A⊗A′) ∼= qM∪N ((C0(RM )⊗A)⊗ (C0(RN )⊗A′)))∆MN

−→ qM ((C0(RM )⊗A)⊗ qN (C0(RN )⊗A′))f⊗g−→ KMB ⊗KNB′ ∼= KM∪NB ⊗B′.

One of course needs to check that this defines a bifunctor of enriched cat-egories. We shall derive this and the main assertion we are after, which isTheorem 9.5, from a general recipe.

9.2. Monoidal product data. Let us be given composition data as in 8.5.Assume further that the C-category D has a symmetric monoidal product inthe sense of Definition 6.9 and Definition 6.10 respectively. A set of monoidalproduct data for D and the composition data consists of natural transformationsof bifunctors

∆MN : FM∪N (A⊗B)→ FM (A)⊗ FN (B);

∇MN : GM (A)⊗GN (B)→ GM∪N (A⊗B)

for any pair of finite subsets M,N ⊂ N with M ∩N = ∅. These functors mustnaturally depend on M and N . Moreover they have to satisfy the followingconditions

(9.2.N) For any triple of pairwise disjoint finite subsets K,L,M,N ⊂ N thenatural transformations must yield commutative diagrams

FK∪L∪M∪N(A⊗B)

// FK∪L(A)⊗ FM∪N (B)

FK∪MFL∪N(A⊗B)

FK∪M (FL(A)⊗ FN (B)) // FKFL(A)⊗ FMFN (B);

GLGK(A) ⊗GNGM (B) //

GL∪N (GK(A)⊗GM (B))

GL∪NGK∪M (A⊗B)

GK∪L(A) ⊗GM∪N (B) // GK∪M∪L∪N (A⊗B);



FK∪MGL∪N(A⊗B) // GL∪NFK∪M (A⊗B)

FK∪M (GL(A)⊗GN (B))

OO

GL∪N (FK(A)⊗ FM (B))

FKGL(A)⊗ FMGN (B) // GLFK(A) ⊗GNFM (B)

OO

(9.2.A) For any three disjoint finite subsets L,M,N ⊂ N and all objectsA,B,C ∈ |D| the following diagram commutes

FL∪M∪N ((A⊗B)⊗ C)

FL∪M∪N (aABC)

∆(L∪M)N// FL∪M (A⊗B)⊗ FN (C)

∆LM⊗1

FL∪M∪N (A⊗ (B ⊗ C))

∆L(M∪N)

FL(A)⊗ FM∪N (B ⊗ C)1⊗∆MN

// FL(A)⊗ FM (B) ⊗ FN (C)

GL(A) ⊗GM (B)⊗GN (C)1⊗∇MN

//

∇LM⊗1

GL(A)⊗GM∪N (B ⊗ C)

∇L(M∪N)

GL∪M∪N (A⊗ (B ⊗ C))

GL∪M∪N (aABC)

GL∪M (A⊗B)⊗GN (C)∇(L∪M)N

// GL∪M∪N ((A⊗B)⊗ C)

(9.2.S) For any pair of disjoint finite sets M,N ⊂ N and all objects A,B ∈ |D|the following diagram commutes

FM∪N (A⊗B)

FM∪N (sAB)

∆MN

// FM (A)⊗ FN (B)

sFM (A),FN (B)

FN∪M (B ⊗A)∆NM

// FN (B)⊗ FM (A)

GMA⊗GNB∇MN

//

sGMA,GNB

GM∪N (A⊗B)

GM∪N (sAB)

GNB ⊗GMA∇NM

// GN∪M (B ⊗A)



(9.2.U) For all M ∈ |I| and A ∈ |D| the following diagrams commute

FM (U ⊗A)∆∅M

//

FM (lA)

U ⊗ FM (A)

lFM (A)

FM (A)= // FM (A)

FM (A⊗ U)

FM (rA)

OO

∆M∅

// FM (A)⊗ U

rFM (A)

OO

U ⊗GM (A)∇∅M

//

lGM (A)

GM (U ⊗A)

GM (lA)

GM (A)= // GM (A)

GM (A) ⊗ U

rGM (A)

OO

∇M∅// GM (A⊗ U)

GM (rA)

OO

Proposition 9.3. Let us be given composition data as in 8.5. Assume furtherthat the C-category D has a symmetric monoidal product, and that we are givenassociated monoidal product data in the sense of 9.2. The following data thendefine a symmetric monoidal CI-category which is an enrichment of D:

(9.3.1) The C-category is D, defined as in Proposition 8.6;(9.3.2) the bifunctor m : D⊗ D→ D is given on objects by (A,A′) 7→ A⊗ A′;

the corresponding morphisms D(A,B)⊗D(A′, B′)→ D(A⊗A′, B⊗B′)for pairs of objects (A,A′) and (B,B′) are given by the maps

D(FMA,GMB)⊗D(FNA′, GNB′)→ D(FM∪N (A⊗A′), GM∪N (B ⊗B′))for disjoint finite sets M,N ⊂ N which send a pair f ∧ g to the com-position

FM∪N (A⊗A′) ∆MN

−→ FM (A)⊗ FN (A′)

f⊗g−→ GM (B)⊗GN (B′)∇MN

−→ GM∪N (B ⊗B′).(9.3.3) the unit is U , the unit of D;(9.3.4) for every triple A,B,C of objects the associativity isomorphism aD

ABC

is the one determined by aDABC(∅) = aDABC : I → D((A⊗B)⊗C,A⊗

(B ⊗ C)) = D∅((A⊗B)⊗ C,A⊗ (B ⊗ C)).(9.3.5) for every object A the left unit isomorphism lDA is the one determined

by lDA(∅) = lDA : I → D(U ⊗A,A) = D∅(U ⊗A,A).(9.3.6) for every object A the right unit isomorphism rD

A is the one determinedby rD

A(∅) = rDA : I → D(A ⊗ U,A) = D∅(A⊗ U,A).

Proof. The commutative diagram displayed in Figure 1 (on the next page)shows the naturality condition (6.3.N) for the bifunctor m; to see commutativ-ity one needs condition (9.2.N). From (9.2.U) it follows that the bifunctor mrespects the unit condition (6.3.U). Similar diagrams show how to verify theassociativity condition (6.9.A) from (9.2.A), the unit condition (6.9.U) from(9.2.U), and the symmetry condition (6.10.S) from (9.2.S).

9.4. KK as an enriched symmetric monoidal category. Before we statethe following theorem recall that the unit of the symmetric monoidal categoryof symmetric pointed spaces is the symmetric pointed space E with E(∅) =S0 = 0,+ and E(M) = + for all nonempty finite subsets M ⊂ N. A map


An

enric

hment

of

KK

-theory

175

FK∪L∪M∪N(A⊗A′)

= // FK∪L∪M∪N (A⊗A′)

FK∪M (A) ⊗ FL∪N(A′) //

FKFM (A)⊗ FLFN (A′)

FK∪L(FM (A)⊗ FN (A′))

oo FK∪LFM∪N (A⊗A′)

oo

FKGM (B)⊗ FLGN (B′)

FK∪L(GM (A)⊗GN (A′))oo // FK∪LGM∪N (B ⊗B′)

GMFK(B)⊗GNFL(B′)

// GM∪N (FK(B)⊗ FL(B′))

GM∪NFK∪L(B ⊗B′)oo

GK∪M (C) ⊗GL∪N(C′)

GKGM (C)⊗GLGN (C′)oo // GK∪L(GM (C)⊗GN (C′)) // GK∪LGM∪N (C ⊗ C′)

GK∪L∪M∪N (C ⊗ C′) = // GK∪L∪M∪N (C ⊗ C′)

Figure 1. The commutative diagram needed to verify the naturality condition in the proof of Proposition 9.3.

Munster

Journ

alofM

ath

ematics

Vol.2

(2009),

143–182


f : E → X from E to a symmetric pointed space X therefore is determinedthrough the point (f(∅))(0) ∈ X(∅).

Theorem 9.5. Let KK be the enrichment of the category C∗-algebras oversymmetric spaces that we introduced in Theorem 8.13. The following data puta symmetric monoidal structure on KK which is an enrichment of symmetricmonoidal structure on the category of C∗-algebras given by the (spatial) tensorproduct

(9.5.2) the bifunctor m : KK ∧KK→ KK is the one introduced in 9.1;(9.5.3) the unit is F;(9.5.4) for every triple of C∗-algebras A,B,C the associativity isomorphism

aKKABC is the one determined by aKK

ABC(∅)(0) = aABC, the associativityisomorphism for the spatial tensor product of C∗-algebras;

(9.5.5) for every object A the left unit isomorphism lKKA is the one determined

by lKKA (∅)(0) = lA, the left unit isomorphism for the spatial tensor

product;(9.5.6) for every object A the right unit isomorphism rKK

A is the one deter-mined by rKK

A (∅)(0) = rA, the right unit isomorphism for the spatialtensor product.

Proof. Let A,B be C∗-algebras, and let M,N be disjoint finite subsets of thenatural numbers N. For these data define ∆MN as in 4.9. On the other hand let∇MN : KMA⊗KB → KM∪N (A⊗B) be the canonical isomorphism which comesfrom the coherence isomorphism of the spatial tensor product. It is straightfor-ward to check that the ∗-homomorphisms ∆MN ,∇MN define monoidal productdata in the sense of 9.2 for the composition data (q•,K•, ˜MN , χMN , ιMN ) thatwe have used in the proof of Theorem 8.13 to define KK. It then follows fromProposition 9.3 that the data given in the theorem define an enrichment of thesymmetric monoidal category of C∗-algebras.

10. The enrichment of the category KK over symmetric spectra

After we have seen (Theorem 9.5) that the category KK is a symmetricmonoidal category which is enriched over the category of symmetric spacesit follows that the endomorphism object of the unit KK(F,F) is a monoid inthe category of symmetric spaces and KK inherits the structure of a sym-metric monoidal category which is enriched over the category of KK(F,F)-modules. To obtain an enrichment over the category of spectra it suffices toturn KK(F,F) into a symmetric ring spectrum, i.e. we need to define a monoidmap from the sphere spectrum S into KK(F,F).

10.1. The ring spectrum K = KK(F,F). Recall that a map of sym-metric sequences η : S → KK(F,F) is determined by maps ηM : SM →Hom(qM (C0(RM )⊗ F),KMF). We define these maps through their adjoints

ηM : S0 → Hom(qM (C0(RM )⊗ F), C0(RM )⊗KMF).



To specify the map ηM we just need to determine the image of the pointwhich is different from the basepoint in S0. This image we define to be the∗-homomorphism

qM (C0(RM )⊗ F)qj→ C0(RM )⊗ F K

j

→ C0(RM )⊗KMF,

where j is the map j : ∅ → M , qj is as defined in (4.7), and Kj is thecomposition

Kj : F ∼= F⊗M ⊗ F e⊗M⊗id−→ KMF,

with e⊗M the M -fold tensor product of the ∗-homomorphism e : F→ K, whichis given by a fixed choice of a rank one projection (cp. 2.1). It is straightforwardto check that these maps define a map of commutative monoids S → K =KK(F,F).

Theorem 10.2. The symmetric monoidal category KK (defined by Theorem8.13 and 9.5) is enriched over the category Sp of symmetric spectra (in fact K-module spectra). The symmetric monoidal Sp-category KK is an enrichmentof the symmetric monoidal category KK by means of the lax-monoidal functorπ0 which associates to a symmetric spectrum its 0-th homotopy group, i.e. thereis a canonical isomorphism

π0(KK(A,B)) ∼= KK (A,B),

which is compatible with composition and the symmetric monoidal structureinduced by the tensor product.

Proof. The first assertion is a consequence of 10.1. It remains to check thatπ0(KK(A,B)) ∼= KK (A,B) for all C∗-algebras A,B. We have π0KK(A,B) =colimn πn(KKn(A,B)) where the structure maps

sn : πn(KKn(A,B))→ πn+1(KKn+1(A,B))

are induced by the structure maps of the spectrum. By construction they fitinto the following diagram

πnKKn(A,B)sn

//

∼=

πn+1KKn+1(A,B)

∼=

π0ΩnKKn(A,B)

∼=

// π0Ωn+1KKn+1(A,B)

∼=

[qn(C0(Rn)⊗A),KnC0(Rn)⊗B] [qn+1(C0(Rn+1)⊗A),Kn+1C0(Rn+1)⊗B]

[qnA,KnB]

cn

OO

// [qn+1A,Kn+1B]

cn+1

OO



Here the maps between the first, second, and third row are induced by adjunc-tion isomorphisms. The homomorphism cn is induced by the map

Hom(qnA,KnB)→ Hom(qn(C0(Rn)⊗A),KnC0(Rn)⊗B)

which maps a ∗-homomorphism f to the composition (idC0(Rn)⊗f)∆∅N (see

4.9 for the definition of ∆∅N ). Via the isomorphisms KK (A,B) ∼= [qnA,KnB]and KK (C0(Rn) ⊗ A,C0(Rn) ⊗ B) ∼= [qn(C0(Rn) ⊗ A),KnC0(Rn) ⊗ B] themap cn corresponds to the Bott periodicity isomorphism (8.10). The map[qnA,KnB] → [qn+1A,Kn+1B] is the stabilization map (2.2). It follows thatπ0KK(A,B) ∼= colimn[qnA,KnB] = KK (A,B) (cp. Section 2).

11. Z/2-graded C∗-algebras

The results of the previous sections deal with ungraded C∗-algebras. In thissections we want to discuss the modifications necessary to deal with Z/2-gradedC∗-algebras.

We recall that a Z/2-grading on a C∗-algebra is a vector space decompo-sition A = A0 ⊕ A1 such that the anti-involution ∗ preserves the Ais, andAi · Aj ⊂ Ai+j , i, j ∈ Z/2. Equivalently, a Z/2-grading on A is just an in-volution of the C∗-algebra A (the +1-eigenspace (resp. −1-eigenspace) of this“grading involution” is the subspace A0 (resp. A1)). Of course every ungradedC∗-algebra A can be interpreted as Z/2-graded C∗-algebra by equipping itwith the trivial grading involution (so that A0 = A, A1 = 0).

Kasparov defined his bivariant KK -groups KK (A,B) not just for C∗-alge-bras A,B, but for Z/2-graded C∗-algebras. It should be emphasized thatKK (A,B) does depend on the gradings of A, B; in other words: in generalKK (A,B) is not isomorphic to KK (Aung , Bung), where Aung, Bung are theC∗-algebras A,B equipped with the trivial grading.

Due to the functoriality of the construction of the Cuntz algebra an in-volution on the C∗-algebra A induces an involution on the Cuntz algebra qA.Hence we may consider the equivariant Cuntz group [qA,K⊗B]Z/2 of homotopyclasses of grading preserving ∗-homomorphisms from qA to K ⊗ B, where Kis the Z/2-graded C∗-algebra of compact operators on a graded Hilbert space,and K ⊗ B is the graded tensor product (which affects the definition of theproduct by setting (a1⊗ b1) · (a2 ⊗ b1) = (−1)|b1||a2|a1b1⊗ a2b2). Cuntz’ argu-ments showing that his group [qA,K ⊗ B] is isomorphic to Kasparov’s groupKK (A,B) generalize to show that there is an isomorphism (natural in botharguments) (cp. [4, (2)])

(11.1) [qA,K ⊗B]Z/2 ∼= KK Z/2(A,B),

where KK Z/2(A,B) is the Z/2-equivariant Kasparov group. Here Z/2 acts onA,B via the grading involutions.

If the C∗-algebras A,B are trivially graded, then the group KK Z/2(A,B)is isomorphic to KK (A,B), but this is not the case in general for Z/2-gradedC∗-algebras. However, these groups are closely related; there is an isomorphism



(natural in both arguments) (cp. [5, Def. 2.3, Prop. 3.8])

(11.2) [q(S ⊗A),K ⊗B]Z/2 ∼= KK (A,B).

Here S is the Z/2-graded C∗-algebra of continuous functions on the real linewhich vanish at∞. The grading involution is induced by the involution R→ R,

x 7→ −x; in other words, S0 consists of the even functions and S1 consists ofthe odd functions.

As in the other sections of this paper we will exclusively work with the Cuntzpicture; consequently, we will take the above isomorphisms as the definitionsof KKZ/2(A,B) resp. KK (A,B).

Next we want to extend the discussion of the previous sections of the functo-rial properties of the KK -groups (the axiomatic characterization a a la Higson,the composition product and the tensor product) from ungraded C∗-algebrasto Z/2-graded C∗-algebras. First we will discuss KKZ/2(A,B), which will turnout to be a straightforward extension of the corresponding results for ungradedalgebras, then we will discuss how to adapt the setup to deal with KK (A,B).

Let C∗Z/2 be the category of Z/2-graded C∗-algebras (i.e., the objects are

C∗-algebras equipped with involutions, and the morphisms are equivariant

∗-homomorphisms). Furnishing KKZ/2(A,B)def= [qA,K⊗B]Z/2 with the struc-

ture of an abelian group as in 2.6, after fixing a Z/2-graded C∗-algebra A weobtain a functor

KK Z/2(A,−) : C∗Z/2 → Ab .

The results of Section 3 generalize to give the following theorem (cp. [5,Thm. 1]).

Theorem 11.3. Let A be a Z/2-graded C∗-algebra.(1) The functor KK Z/2(A,−) : C∗Z/2 → Ab is homotopy invariant, stable

and split exact.(2) If F : C∗Z/2 → Ab is a homotopy invariant, stable and split exact func-

tor, and x ∈ F (A), then there is a unique natural transformationα : KK Z/2(A,−)→ F with αA(1A) = x.

We can define “composition products” and “tensor products” for the Z/2-equivariant KK -groups, by the same formulas as in the nonequivariant case.As in the nonequivariant case, Theorem 11.3 implies that these products areunique. Moreover, we obtain a symmetric monoidal category KK Z/2 whoseobjects are C∗-algebras with involutions, and whose set of morphisms from Ato B is KK Z/2(A,B).

Now we will derive an axiomatic characterization of KK (A,B) for Z/2-equi-variant C∗-algebras A,B which is analogous to the axiomatic characterizationof KKZ/2(A,B) in Theorem 11.3. The idea is to replace the category C∗Z/2 by

the following category.

11.4. The category C∗Z/2. The objects of C∗Z/2 are Z/2-graded C∗-algebras.

A morphism from A to B is a Z/2-equivariant ∗-homomorphism from S ⊗ A



to B. The composition

C∗Z/2(A,B)× C∗Z/2(B,C)→ C∗Z/2(A,C)

sends a pair of morphisms f : S ⊗ A → B, g : S ⊗ B → C to the morphismgiven by

(11.5) S ⊗A ∆⊗1A→ S ⊗ S ⊗A 1 bS⊗f→ S ⊗B g→ C.

Here ∆: S ⊗ S → S is the Z/2-equivariant ∗-homomorphism dual to the ad-

dition map R × R → R if we identify S ⊗ S with the space of functions oftwo anti-commuting variables x, y. To illustrate what is meant, consider the

function f(x) = e−x2 ∈ S. Then ∆(f) is a function of two variables, say x and

y given by ∆(f) = f(x+ y). To interpret f(x+ y) as an element of S ⊗ S, weexpand f(x+ y) in terms of x and y.

∆(ex2

) = e(x+y)2

= ex2+xy+yx+y2

= ex2

ey2 ∈ S ⊗ S.

Here the third equality holds, since the variables x, y are assumed to anti-commute. Similarly, we have:

∆(xex2

) = (x+ y)e(x+y)2

= (x+ y)ex2

ey2

= (xex2

)ey2

+ ex2

(yey2

) ∈ S ⊗ S.

We note that the C∗-algebra S is generated by ex2

and xex2

. So we could havedefined ∆ by the above equations. Thinking of ∆ as induced by addition makesit obvious that the “coproduct” ∆ is Z/2-equivariant (since +: R × R → Ris) and the ∆ is coassociative, while from the other point of view this needsa little calculation to check. Coassociativity of ∆ implies associativity of thecomposition defined above.

11.6. The category KKZ/2. The category KKZ/2 is built out of the category

KK Z/2 the same way the category C∗Z/2 is built from the category C∗Z/2; i.e.,

• The objects of KK Z/2 are the Z/2-graded C∗-algebras;

• KK Z/2(A,B)def= KKZ/2(S⊗A,B); i.e., a morphism from A to B in the

category KK Z/2 is just a morphism from S ⊗ A to B in the categoryKK Z/2.

• the composition of a morphism f ∈ KKZ/2(A,B) with a morphism

g ∈ KK Z/2(B,C) is given by the formula 11.5 with the only differencethat now these arrows have to be interpreted as morphisms in thecategory KK Z/2 instead of as morphisms in C∗Z/2 (∆ is interpreted as

morphism in KK Z/2 by means to the obvious functor C∗Z/2 → KK Z/2).

Passing from Z/2-equivariant ∗-homomorphisms S ⊗ A → B to elements of

KK Z/2(S ⊗A,B) then defines a functor

C∗Z/2 → KK Z/2,



and, after fixing a Z/2-graded C∗-algebra A, a functor

KK Z/2(A,−) : C∗Z/2 → Ab .

We note that according to the isomorphism 11.2, we have KK Z/2(A,B) ∼=KK (A,B) for C∗-algebras A, B.

Theorem 11.3 then implies the following result.

Theorem 11.7. Let A be a Z/2-graded C∗-algebra.

(1) The functor KK Z/2(A,−) : C∗Z/2 → Ab is homotopy invariant, stable

and split exact.

(2) If F : C∗Z/2 → Ab is a homotopy invariant, stable and split exact func-

tor, and x ∈ F (A), then there is a unique natural transformation

α : KK (A,−)→ F with αA(1A) = x.

11.8. Enrichments of the categories KK Z/2 and KK Z/2 over the cate-gory of symmetric spectra. Since the Cuntz stabilization isomorphism ([5,Theorem 2.4]) and Bott periodicity (in the sense of Theorem 8.9) also hold forthe Z/2-graded setting one obtains completely analogous to the treatment ofthe ungraded setting an enrichment KKZ/2 of KK Z/2 over the category of sym-metric spectra. For a pair A,B of Z/2-graded C∗-algebras the correspondingsymmetric space KKZ/2(A,B) is given by

KKZ/2(A,B) : M 7→ HomZ/2(qM (C0(RM )⊗A),K ⊗B).

For the category KK Z/2 one obtains an enrichment over symmetric spectra,if we define the morphism spectra by

KKZ/2(A,B) : M 7→ HomZ/2(qM (S ⊗ C0(RM )⊗A),K ⊗B).

On the formal level the treatment of this case is completely analogous to oneabove; however one has to work with the morphisms and the tensor product of

the category C∗Z/2 instead of honest ∗-homomorphisms and the standard tensor

product.

References


[2] F. Borceux, Handbook of categorical algebra. 2, Cambridge Univ. Press, Cambridge,1994. MR1313497 (96g:18001b)

[3] J. Cuntz, A new look at KK-theory, K-Theory 1 (1987), no. 1, 31–51. MR0899916(89a:46142)

[4] U. Haag, Some algebraic features of Z2-graded KK -theory, K-Theory 13 (1998), no. 1,81–108. MR1610896 (99f:19003)

[5] U. Haag, On Z/2Z-graded KK-theory and its relation with the graded Ext-functor, J.Operator Theory 42 (1999), no. 1, 3–36. MR1694805 (2000j:19004)

[6] N. Higson, A characterization of KK-theory, Pacific J. Math. 126 (1987), no. 2, 253–276. MR0869779 (88a:46083)

[7] M. Hovey, B. Shipley and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000),no. 1, 149–208. MR1695653 (2000h:55016)



[8] G. G. Kasparov, The operator K-functor and extensions of C∗-algebras, Izv. Akad.Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719. MR0582160 (81m:58075)

[9] G. M. Kelly, On MacLane’s conditions for coherence of natural associativities, commu-tativities, etc., J. Algebra 1 (1964), 397–402. MR0182649 (32 #132)

[10] M. Schlichting, Delacage de la k-theorie des categories exactes et k-groupes negatifs.These, Universite Paris 7 Denis Diderot (2000).

[11] N. E. Wegge-Olsen, K-theory and C∗-algebras, Oxford Univ. Press, New York, 1993.MR1222415 (95c:46116)

[12] J. Weidner and R. Zekri, An equivariant version of the Cuntz construction, J. OperatorTheory 27 (1992), no. 1, 3–15. MR1241113 (94i:46092)

Received April 22, 2009; accepted June 16, 2009

Michael JoachimWestfalische Wilhelms-Universitat Munster, Mathematisches InstitutEinsteinstr. 62, D-48149 Munster, GermanyE-mail: [email protected]

URL: http://www.math.uni-muenster.de/u/joachim

Stephan StolzUniversity of Notre Dame, Department of MathematicsNotre Dame, IN 46556, USAE-mail: [email protected]

URL: http://www.nd.edu/~stolz/




Skew Hopf algebras, irreducible extensions and

the Π-method

Lars Kadison



Abstract. To a depth two extension A|B, we associate the dual bialgebroids S := End BAB

and T := (A ⊗B A)B over the centralizer R = CA(B). In a set-up which is quite common,where R is a subalgebra of B, two nondegenerate pairings of S and T will define an anti-automorphism τ of the algebra S. Making use of a two-sided depth two structure, we showthat τ is an antipode and S is a Hopf algebroid of a type we call skew Hopf algebra. Afinal section discusses how τ and the nondegenerate pairings generalize to modules via theπ-method for depth two.

1. Introduction

For reasons of symmetry in representation theory, given a bialgebra or bial-gebroid one would like to expose the presence of an antipode. In the re-construction of Hopf algebras and weak Hopf algebras in subfactor theory akey idea in the definition of antipode is to make use of the existence of twonondegenerate pairings of dual bialgebras. In terms of a depth two, finite in-dex subfactor N ⊆ M with trivial relative commutant, its basic constructionM ⊆M1, and another one above, M1 ⊆M2, there are conditional expectationsEM : M1 → M and EM1 : M2 → M1. In addition to the Jones projectionse1 ∈ M1 and e2 ∈ M2, the two relative commutants that are paired non-degenerately are in ordinary algebraic centralizer notation C = CM1(N) andV = CM2 (M). The antipode τ : V → V is then defined as the “difference” oftwo such pairings:

(1) EMEM1(ve1e2c) = EMEM1 (ce2e1τ(v))

for c ∈ C and v ∈ V : see for example, [8, 17, 19] for the details of why thisformula works.

The duality method for defining antipode has been lying dormant in recentgeneralizations of depth two to algebras and rings and actions of bialgebroids on

184 Lars Kadison

these. For example, antipodes have been defined recently in the case of a Frobe-nius extension A|B as the restriction of a standard anti-isomorphism of the leftand right endomorphism rings EndAB → EndBA to an anti-automorphism ofthe subring of bimodule endomorphisms S = EndBAB: on depth two Frobe-nius extension this defines the antipode or its inverse in [2] in a dual way onboth S and T ; it also necessitates a revision of the definition of the notion ofHopf algebroid using the notions of left and right bialgebroid. Antipodes havealso been defined from geometric ideas of Lu [16] for H-separable extensions[10], extensions of Kanzaki separable algebras [12] and Hopf-Galois extensions[11], and from group theory in [14] for pseudo-Galois extensions.

In this paper we define antipode as the difference of two nondegeneratepairings for a special extension A|B where the centralizer R is a subalgebra ofthe smaller algebra B, which we call irreducible extension. In this case, thetwo hom-groups, the left and right R-duals of the bialgebroid T , coincide. Theplan to find an antipode from this identity and satisfying the several axiomsof a Hopf algebroid works not so much because of any Frobenius structure(as assumed previously) but on a two-sided depth two structure as shown insections 3 and 4 below. We recall that the dual bialgebroids S and T dependonly a one-sided depth two structure [12], but at two stages in this paper(using both nondegenerate pairing and the proposition in section 3) we requirea two-sided depth two structure. The Frobenius extension hypothesis avoidedin this paper, makes one-sided depth two extensions two-sided [9, 6.4]. TheHopf algebroid structure we obtain on S is very nearly a Hopf algebra which isfinite projective over a commutative base ring: we discuss its properties afterTheorem 4.1 and designate as skew Hopf algebras such a Hopf algebroid. Weend with a discussion of how τ and the nondegenerate pairings generalize tomodules via the π-method for depth two. A certain mapping between cochaincomplexes formed from the left- and right-handed π-methods is shown to benullhomotopic.

2. Preliminaries on depth two extensions

Let B be a unital subalgebra of A, an associative noncommutative algebrawith unit over a commutative ground ring K. The algebra extension A|B isdepth two if there is a positive integer N and bimodule P such that

(2) A⊗B A⊕ P ∼= AN

as natural B-A (left D2) and A-B-bimodules (right D2) [9]. For an A-A-bimodule M , denote the subgroup of B-central elements in M by

MB := m ∈M |∀b ∈ B, bm = mb.

Equivalently, the algebra extension A|B is depth two if there are elements

βi ∈ S := End BAB , ti ∈ T := (A⊗B A)B


Skew Hopf algebras and irreducible extensions 185

(called a left D2 quasibasis) such that simple tensors of A⊗BA may be writtenas (a, a′ ∈ A)

(3) a⊗ a′ =N∑

i=1

tiβi(a)a′,

and similarly, there are elements (of a right D2 quasibasis) γj ∈ S, uj ∈ Tsuch that

(4) a⊗ a′ =∑

j

aγj(a′)uj.

For example, given a right D2 quasibasis, define a split A-B-epimorphismAN → A⊗B A by

(a1, . . . , aN ) 7−→N∑

j=1

ajuj

which is split by the A-B-monomorphism A ⊗B A → AN given by x ⊗By 7→ (xγ1(y), . . . , xγN (y)). Conversely, given a split epi AN → A ⊗B A,

we obtain mappings∑N

i=1 fi gi = idA⊗BA where gi ∈ Hom(A ⊗B A,A)and fi ∈ Hom(A,A ⊗B A); but there are somewhat obvious isomorphismsHom (A,A ⊗B A) ∼= T and Hom (A ⊗B A,A) ∼= S in either case of A-B- orB-A-bimodule homomorphisms. We fix the notations for both right and leftD2 quasibases throughout this paper.

For example, an H-separable extension A|B is of depth two since the con-dition above on the tensor-square holds even more strongly as natural A-A-bimodules. Another example: A a f.g. projective algebra over commutativeground ring B, since left or right D2 quasibases are easily constructed from adual basis. As a third class of examples, consider a Hopf-Galois extension A|Bwith n-dimensional Hopf k-algebra H [15]. Recall that H acts from the left onA with subalgebra of invariants B, induces a dual right coaction A→ A⊗kH∗,a 7→ a(0) ⊗ a(1), and Galois isomorphism β : A ⊗B A

∼=−→ A ⊗k H∗ given byβ(a⊗ a′) = aa′(0)⊗ a′(1) , which is an A-B-bimodule, right H∗-comodule mor-phism. It follows that A ⊗B A ∼= ⊕nA as A-B-bimodules; as B-A-bimodulesthere is a similar isomorphism by making use of the alternative Galois iso-morphism β′ given by β′(a ⊗ a′) = a(0)a

′ ⊗ a(1). The paper [13] extends thedefinition above of depth two to include the case where the tensor-square ofA|B is isomorphic to any direct sum of A with itself (not necessarily a finitedirect sum as in eq. 2); thus any Hopf-Galois extension is depth two in this ex-tended sense. However, this theory does not have a theory of dual bialgebroidscongenial for the results in this paper, and we shall not make use of it.

The papers [9, 10] defined dual bialgebroids with action and smash productstructure within the endomorphism ring tower construction above a depthtwo ring extension A|B. In more detail, if R denotes the centralizer of Bin A, a left R-bialgebroid structure on S is given by the composition ringstructure on S with source and target mappings corresponding to the leftregular representation λ : R→ S and right regular representation ρ : Rop → S,


186 Lars Kadison

respectively. Since these commute (λrρs = ρsλr for every r, s ∈ R), we mayinduce an R-bimodule structure on S solely from the left by

r · α · s := λrρsα = rα(−)s.

Now an R-coring structure (S,∆, ε) is given by

(5) ∆(α) :=∑

i

α(−t1i )t2i ⊗R βi

for every α ∈ S, denoting ti = t1i ⊗B t2i ∈ T by suppressing a possible summa-tion, and

(6) ε(α) = α(1)

satisfying the additional axioms of a bialgebroid (cp. appendix), such as mul-tiplicativity of ∆ and a condition that makes sense of this requirement. Wehave the equivalent formula for the coproduct [9, Thm. 4.1]:

(7) ∆(α) :=∑

j

γj ⊗R u1jα(u2

j−)

For a depth two extension,

(8) S ⊗R S∼=−→ Hom(BA⊗B AB ,BAB), α⊗R β 7−→ (x ⊗B y 7→ α(x)β(y))

(or α⊗R β 7→ α ∪ β thinking of Hochschild cochain cup product) with inverseprovided by F 7→∑

i F (−⊗ t1i )t2i ⊗R βi. Under this identification, the formulafor the coproduct ∆ : S → S ⊗R S becomes

(9) α(1)(x)α(2)(y) = α(xy)

using either equation. (This isomorphism extends to any number of S tensoredwith itself over R, the so-called Amitsur complex of R-coring S with grouplikeelement idA, which is then isomorphic as differential graded algebras to relativeHochschild complex of A over B with values in A.)

The left action of S on A given by evaluation, α ⊲ a = α(a), has invariantsubalgebra (of elements a ∈ A such that α ⊲ a = ε(α)a) equal precisely to B ifthe natural module AB is balanced [9]. This action is measuring by eq. (9).

The smash product A⋊S, which is A⊗RS as abelian groups with associativemultiplication given by

(10) (x⋊ α)(y ⋊ β) = x(α(1) ⊲ y) ⋊ α(2)β,

is isomorphic as rings to EndAB via a⊗R α 7→ λaα [9].In general T = (A ⊗B A)B has a unital ring structure induced from T ∼=

End A(A⊗BA)A via F 7→ F (1⊗ 1), which is given by

(11) tu = u1t1 ⊗ t2u2

for each t, u ∈ T . There are obvious commuting homomorphisms of R and Rop

into T given by r 7→ 1 ⊗ r and s 7→ s ⊗ 1, respectively. From the right, thesetwo source and target mappings induce the R-R-bimodule structure RTR givenby

r · t · s = (t1 ⊗ t2)(r ⊗ s) = rt1 ⊗ t2s,



the ordinary bimodule structure on a tensor product.There is a right R-bialgebroid structure on T with coring structure (T,∆, ε)

given by the two equivalent formulas:

∆(t) =∑

i

ti ⊗R (βi(t1)⊗B t2) =

∑

j

(t1 ⊗B γj(t2))⊗R uj(12)

ε(t) = t1t2(13)

By [9, Thm. 5.2] ∆ is multiplicative and the other axioms of a right bialgebroidare satisfied.

As an example of S and T , consider the Hopf-Galois extension A|B of k-algebras introduced above. Since β is an A-B-isomorphism, we may computethat T ∼= R⊗k H∗ via β, which induces a smash product structure on R⊗H∗relative to the Miyashta-Ulbrich action of H∗ on R from the right. The well-known isomorphism EndAB ∼= A ⋊ H via a ⋊ h 7→ λ(a)(h ⊲ ·) restricts toS ∼= R ⋊ H , i.e., S is a smash product of R with H via the restriction of theleft action of H to R. In both cases, the R-coring structures are the trivialones induced from the coalgebras H and H∗.

There is a right action of T on E := End BA given by f ⊳ t = t1f(t2−) forf ∈ E . This is a measuring action by Eq. (3) since

(f ⊳ t(1)) (g ⊳ t(2)) =∑

i

t1i f(t2iβi(t1)g(t2−)) = fg ⊳ t.

The subring of invariants in E is ρ(A) [9]. Also, in analogy with EndAB ∼=A⋊S, the smash product ring T ⋉E is isomorphic to End AA⊗BA via Ψ givenby

(14) Ψ(t⊗ f)(a⊗ a′) = at1 ⊗B t2f(a′).

Sweedler [18] defines left and right R-dual rings of an R-coring. In the caseof a left R-bialgebroid H with HR and RH finitely generated projective, suchas (S, λ, ρ,∆, ε) above, the left and right Sweedler R-dual rings are extendedto right bialgebroids H∗ and ∗H in [9]. For example, H∗ has a natural nonde-generate pairing with H denoted by 〈h∗, h〉 ∈ R for h∗ ∈ H∗, h ∈ H . Then theR-bimodule structure on H∗, multiplication, and comultiplication are given

below, respectively, where Rs→ H

t← Rop denotes the commuting morphismset-up of the bialgebroid H :

〈r · h∗ · r′, h〉 := r〈h∗, ht(r′)〉(15)

〈h∗g∗, h〉 := 〈g∗, 〈h∗, h(1)〉 · h(2)〉(16)

〈h∗, hh′〉 := 〈h∗(1) · 〈h∗(2), h′〉, h〉(17)

Of course, the unit of H∗ is εH while the counit on H∗ is ε(h∗) = 〈h∗, 1H〉.Eq. (16) is the formula for multiplication [18, 3.2(b)].

There are similar formulas for the right bialgebroid structure on the leftR-dual ∗H : see [9, 2.6]. In the particular case of the left bialgebroid S of adepth two ring extension, it turns out that S is isomorphic as R-bialgebroids


188 Lars Kadison

to both R-duals, T ∗ and ∗T via two nondegenerate pairings, one of which isgiven by (α ∈ S, t ∈ T ):

(18) 〈α, t〉 = α(t1)t2 ∈ RThis induces an isomorphism of left R-modules S → Hom(TR, RR) via α 7→〈α,−〉 with inverse

φ 7→∑

i

φ(ti)βi.

Significantly, there is another nondegenerate pairing of S and T for left andright D2 extensions given by

(19) [t, α] = t1α(t2)

[9, 5.3]. This induces an isomorphism of right R-modules S → Hom(RT,RR)given by α 7→ [−, α], with inverse given from a right D2 quasibasis by

ψ 7→∑

j

γj(−)ψ(uj).

3. Irreducible extensions

We define a class of depth two extension where we may readily exploit thetwo nondegenerate pairings just given in eqs. (18) and (19). We say thatan algebra extension A|B is irreducible if it is depth two and its centralizerCA(B) = R is a subalgebra of B, so R ⊆ B. Then R is a commutativesubalgebra, since rs = sr for all s, r ∈ R follows from noting for instance thatr ∈ B and s ∈ AB .

This set-up is quite common. For example, irreducible depth two subfactorsare irreducible in our sense since the centralizer is one-dimensional over thecomplex numbers [8, 17]. A second example: Taft’s Hopf algebras includingSweedler’s 4-dimensional Hopf algebra, which are generated by a grouplikeelement g and a skew-primitive element x, over the commutative Frobeniussubalgebra B generated by the element x: this extension satisfies B = R andis depth two (in fact strongly graded, therefore Hopf-Galois) [6, 7]. A thirdexample is the extension C ⊂ H , the complex numbers as a subring in the realquaternions.

Another type of example of irreducible extension is the H-separable exten-sion of full n× n matrix algebra over the triangular matrix subalgebra, whichof course has trivial centralizer. (If we pass to infinite dimensional matrices offinite type, a version of this example shows that Cuadra’s result for separableHopf-Galois extension [4] does not extend to separable, depth two extensions;namely, an example of a infinitely generated H-separable extension.) Finally,note that an intermediate ring B in an irreducible extension A |C is irreducibleif A |B is D2, since AB ⊆ AC ⊆ C ⊆ B.

For an irreducible extension A|B with the construction T = (A⊗B A)B, wenote that

(20) Hom (TR, RR) = Hom (RT,RR).



This follows from R ⊆ B and commutativity in R, for given φ ∈ Hom(TR, RR),t ∈ T, r ∈ R:

φ(rt) = φ(tr) = φ(t)r = rφ(t),

and a similar computation showing ∗T ⊆ T ∗ using left and right R-dual nota-tion. In case A|B is (two-sided) depth two and irreducible, the two nondegen-erate pairings (α, β ∈ S)

S∼=−→ Hom(TR, RR), α 7−→ 〈α,−〉

and

S∼=−→ Hom(RT,RR), β 7−→ [−, β]

induce a bijection τ of S with itself completing a commutative triangle withthese two mappings. Then define

(21) τ : S → S, 〈α, t〉 = [t, τ(α)]

for all t ∈ T and α ∈ S. We also make use of the notation ατ = τ(α), forwhich the last equation becomes

(22) α(t1)t2 = t1ατ (t2).

Notice that this approach will not work on the two nondegenerate pairingsT → Hom(S,R), to define a self-bijection on T , unless we assume that Rcoincides with the center of A.

Lemma 3.1. The mapping τ : S → S is an anti-automorphism of S satisfyingτ(ρr) = λr, τ(λr) = ρr for r ∈ R and ατ (1) = α(1).

Proof. It is clear that τ is linear and bijective. We note that for t = t1⊗B t2 ∈T , α, β ∈ S,

t1βτατ (t2) = β(t1)ατ (t2) = αβ(t1)t2 = t1(αβ)τ (t2)

since t1 ⊗B ατ (t2) and β(t1)⊗B t2 both are in T . Then [t, βτατ ] = [t, (αβ)τ ],so by nondegeneracy of this pairing, τ is an anti-automorphism of S.

We also check that

[t, ρτr ] = 〈ρr, t〉 = t1rt2 = [t, λr]

whence ρτr = λr. Since R ⊂ B and T = (A⊗B A)B , we note that

[t, ρr] = t1t2r = rt1t2 = 〈λr, t〉 = [t, λτr ]

whence λτr = ρr for each r ∈ R.Finally, with 1T = 1⊗B 1, the equality α(1) = ατ (1) follows from 〈α, 1T 〉 =

[1T , ατ ].

Proposition 3.2. Suppose A|B is an irreducible extension with left D2 qua-sibasis ti ∈ T, βi ∈ S and right D2 quasibasis uj ∈ T, γj ∈ S. Then ti, β

τi is a

right D2 quasibasis and uj, γτj is a left D2 quasibasis.


190 Lars Kadison

Proof. Note that for t ∈ T , a special instance of eq. (3) yields

t =∑

i

t1i ⊗B t2iβi(t1)t2 =∑

i

t1βτi (t2)t1i ⊗B t2i ,

since βi(t1)t2 ∈ R ⊆ B. Now recall that A⊗RT ∼= A⊗BA via a⊗Rt 7→ at1⊗Bt2

since A|B is right D2 (for an inverse is given by x⊗B y 7→∑j xγj(y)⊗R uj).

Note that

A⊗B A −→ A⊗R T, x⊗B y 7−→∑

i

xβτi (y)⊗R ti

is a left inverse of A⊗R T → A⊗B A, a⊗R t 7→ at1 ⊗B t2 since∑

i

at1βτi (t2)⊗R ti = a⊗R∑

i

t1βτi (t2)ti = a⊗R t.

Hence, it is also a right inverse, so

(23) x⊗B y =∑

i

xβτi (y)ti

for all x, y ∈ A, which shows that ti ∈ T, βτi ∈ S is a right D2 quasibasis.The argument that uj , γ

τj is a left D2 quasibasis is very similar.

4. The Hopf algebroid S

We are now in a position to show that the anti-automorphism τ on S, definedin eq. (22), is an antipode satisfying the axioms of Bohm-Szlachanyi [2]. Inorder for S to be a Hopf algebroid in the sense of Lu, we need one additionalrequirement, e.g. that R be a separable K-algebra, in order that we may finda section of the canonical epi S ⊗K S → S ⊗R S.

Theorem 4.1. If A|B is an irreducible extension, then the anti-automorphismτ on S = EndBAB is an antipode and S is a Hopf algebroid.

Proof. We check that the axioms of a Hopf algebroid are satisfied, axiomsgiven in for example [9, 8.7] and repeated in an appendix below. Define a rightbialgebroid structure on S over Rop = R by choosing target map tR = λ andsource map sR = ρ, whence the R-bimodule structure on S becomes

r · α · s = αρsλr = α(r?s) = rα(?)s,

the usual structure on S introduced above (since R ⊆ B). Then the leftbialgebroid structure (S,R,∆, ε) introduced above in eqs. (5), (6) and (7) isalso a right bialgebroid structure. In other words, the axioms (1) and (2) in [9,8.7] are satisfied by noting that sL = tR, tL = sR, and taking ∆L = ∆R andεL = εR. We check that also the axiom of a right bialgebroid, sR(r)α(1) ⊗Rα(2) = α(1) ⊗R tR(r)α(2) is satisfied since ρrα(1)(x)α(2)(y) = α(1)(x)λrα(2)(y)(x, y ∈ A) in the identification S⊗R S ∼= Hom(BA⊗B AB,BAB) in eq. (8). Inaddition, the axiom ε(tR(ε(α))β) = ε(αβ) = ε(sR(ε(α)β) (α, β ∈ S) is satisfiedsince

ε(λε(α)β) = α(1)β(1) = α(β(1)) = ε(αβ)



which equals β(1)α(1) = ε(ρε(α)β), where we use α(1) ∈ R ⊆ B and R iscommutative.

We proceed to axiom (i):

τ(αtR(r)) = τ(λr)τ(α) = sR(r)τ(α)

by the lemma, and

τ(tL(r)α) = τ(α)τ(ρr) = τ(α)sL(r)

for all r ∈ R,α ∈ S.Finally, axiom (ii) is satisfied since by eq. (5)

α(1) τ(α(2)) =∑

i

α(βτi (−)t1i )t2i = λα(1) = sL(ε(α))

by the proposition, and by lemma,

τ(α(1)) α(2) =∑

i

τ(ρt2iαρt1i ) βi =∑

i

λt1iατλt2i βi

= ρατ (1) = sR(ε(α)).

Thus S and τ form a Hopf algebroid.

For example, the Hopf algebroid structure on S coincides with that in [10]should the extension be H-separable as well as irreducible, since τ exchangesλr and ρr.

Note that Hopf algebroid (S, τ) satisfies properties close to a Hopf algebra,among them:

(1) τ(r · α) = τ(α) · r and τ(α · r) = r · τ(α) for all r ∈ R,α ∈ S;(2) ε(αβ) = ε(α)ε(β) for all α, β ∈ S;(3) α(1)τ(α(2)) = ε(α) · 1S for all α ∈ S;(4) τ(α(1))α(2) = 1S · ε(α) for all α ∈ S;(5) ετ = ε;(6) τ is an “anti-coalgebra homomorphism.”

Also, S is finite projective over the base ring R, which is commutative. How-ever, such basic algebraic properties of a Hopf algebra as r · 1 = 1 · r and(α · r)β = α(r · β) are suspended for S (unless R coincides with the center ofA). We propose to call a Hopf algebroid with equal right and left bialgebroidstructures over a commutative base ring, possessing an anti-automorphism ex-changing source and target, both mappings with image in the center, and sat-isfying the properties enumerated directly above, a skew Hopf algebra. Giventhe general nature of the example S, we would expect that skew Hopf algebrasare quite common occurrences.

Corollary 4.2. Suppose A|B is an irreducible extension and R is a separableK-algebra. Then (S, τ) is a Hopf algebroid in the sense of Lu.

Proof. Let e = e1 ⊗K e2 be a separability element for R. Define a sectionη : S ⊗R S → S ⊗K S of the canonical epi S ⊗K S → S ⊗R S by η(α⊗R β) =


192 Lars Kadison

α · e1 ⊗K e2 · β, since e1e2 = 1 and re = er for r ∈ R. Then the axiomµ(id⊗ τ)η∆ = sL ε follows from

[µ(id⊗ τ)η∆(α), u] =∑

i

u1α(βiτ (u2e2)t1i )t

2i e

1 = u1α(1)u2 = [λα(1), u]

by the proposition and since R ⊆ B, e2e1 = 1. The other axioms follow as inthe proof of the theorem.

Of course, like the inverse in group theory, antipodes are important to therepresentation theory of a bialgebroid. For example, we can now define a rightS-module algebra structure on A from the left structure by a ⊳ α = ατ ⊲ a,which satisfies the measuring rule (xy) ⊳ α = (x ⊳ α(2))(y ⊳ α(1)) for x, y ∈ Aand α ∈ S.

The antipode on S will not dualize readily to an antipode on T withoutthe duality properties discussed in [3], such as S possessing a nondegenerateintegral element (in Hom (BAB ,BBB) such as a Frobenius homomorphism).However, if R = Z(A), as mentioned above an antipode τT is definable in thesame way as τ = τS .

The computations in this section expose the hypotheses that are necessaryfor the antipode defined in [8, 4.4]. The correspondence of the theory in thissection with that in [8] is discussed in [9, 8.9], and depends on the equation fora depth two Frobenius extension A|B with trivial one-dimensional centralizer:

(24) [t, α] = EMEM1(ψ(t)e1e2φ(α))

for certain anti-isomorphisms ψ : T → CM2 (M) defined in [9, 8.2] and φ : S →CM1 (N) defined in [9, 8.4].

5. The π-method for depth two extensions

In this section we extend Doi and Takeuchi’s π-method for Hopf-Galoisextensions [5] to D2 extensions. Then we extend the antipode in the previ-ous section to a certain bimodule hom-group for irreducible extensions. Wepoint out that the π-method yields a nullhomotopic mapping between relativeHochschild cochains with coefficients for the irreducible extension A |B and acertain Hochschild cohomology theory with coefficients for the R-coring T .

Suppose A |B is an rD2 extension, and AM is a module. Again let T =(A ⊗B A)B , the right bialgebroid over the centralizer R = AB. Let uj ∈ Tand γj ∈ S be rD2 quasibases. Recall from [12] the coaction ρT : A→ A⊗R Twhich makes A into a right comodule algebra: in Sweedler notation this isgiven by

(25) a(0) ⊗R a(1) =∑

j

γj(a)⊗R uj

Clearly, b(0) ⊗ b(1) = b⊗ 1T if b ∈ B.



Proposition 5.1. The mapping πL : Hom (RT,RM) −→ Hom(BA,BM)given by

(26) πL(f)(a) = a(0)f(a(1))

is a left B-linear, R-linear isomorphism.

Proof. The inverse to πL is given by

(27) π−1L (g)(t) = t1g(t2)

for g ∈ Hom(BA,BM). We note that

t1πL(f)(t2) =∑

j

t1γj(t2)f(uj) = f(t)

for f ∈ Hom(RT,RM), t ∈ T , since t1γj(t2) ∈ AB = R and

∑j t

1γj(t2)uj = t.

We also note that ∑

j

γj(a)u1jg(u

2j) = g(a)

for a ∈ A, g ∈ Hom(BA,BM) by eq. (4). Hence, π−1L is indeed the inverse of

πL.The mapping πL is left B-linear, since

πL(bf)(a) = a(0)bf(a(1)) = π(f)(ab).

Note that π−1L is left R-linear since

π−1L (rg)(t) = t1(rg)(t2) = t1g(t2r) = π−1

L (g)(tr).

Similarly, if A |B is ℓD2, NA is a module and βi ∈ S, ti ∈ T are ℓD2quasibases, then we have the right module dual of the proposition:

(28) πR : Hom (TR, NR)∼=−→ Hom(AB , NB), πR(h)(a) =

∑

i

h(ti)βi(a),

which has inverse mapping given for g ∈ Hom(AB , NB), t ∈ T by

(29) π−1R (g)(t) = g(t1)t2

Suppose that APA is a bimodule. Note that the B-central subgroup PB isa natural (R,R)-bimodule, since R and B commute.

First note that πL : Hom (RT,RP )∼=−→ Hom(BA,BP ) restricts to

(30) πL : Hom (RT,RPB)

∼=−→ Hom(BAB ,BPB)

Similarly πR above restricts to

(31) πR : Hom (TR, PBR )

∼=−→ Hom(BAB ,BPB).

Note that the inverses are given by

(32) π−1R (h)(u) = h(u1)u2, π−1

L (h)(u) = u1h(u2)

for h ∈ Hom(BAB,BPB), u ∈ T . Of course if P = A these restricted mappingsrecover the isomorphisms Hom (TR, RR) ∼= S ∼= Hom(RT,RR) below eq. (18).


194 Lars Kadison

Now suppose that the ring extension A |B is irreducible. The antipode inthe previous sections then extends to a bijection of the hom-group Hom (BAB ,

BPB) onto itself as follows. Since R ⊆ B, it follows that Hom (TR, PBR ) =

Hom (RT,RPB). Then define the mapping

σP = πL π−1R : Hom (BAB ,BPB)

∼=−→ Hom(BAB ,BPB)

which is then given by

(33) σP (α)(a) = πL(π−1R (α))(a) =

∑

j

γj(a)α(u1j )u

2j

for α ∈ Hom(BAB,BPB). Now let ασ = σP (α). Then for all t ∈ T ,

(34) α(t1)t2 = t1ασ(t2),

which follows from the following short computation:

t1σP (α)(t2) =∑

j

t1γj(t2)α(u1

j )u2j = α(t1)t2,

since t1γj(t2) ∈ R ⊆ B and eq. (4). A glance at eq. (22) shows that σA = τ ,

the antipode of S defined in previous sections. The proof of the proposition issimilar to previously and therefore omitted.

Proposition 5.2. Suppose A |B is an irreducible extension and APA is abimodule. Define two pairings of Hom(BAB,BPB) and T with values in PB by〈α, t〉 = α(t1)t2 and [u, β] = u1β(u2). Then the two pairings are nondegeneratew.r.t. α, β ∈ Hom(BAB ,BPB). Moreover, the mapping α 7→ ασ defined by〈α, t〉 = [t, ασ] is a bijection satisfying σ±1(λp) = ρp for each p ∈ P , andασ(1A) = α(1A).

5.1. Remark on relative Hochschild cohomology with coefficients. Weremark below on how the π-method leads to a nullhomotopic mapping betweencertain Hochschild cohomology theories. Continuing the notation just above,note that Hom (BAB,BPB) is the first relative Hochschild cochain group ofA |B with coefficients in a bimodule APA, denoted by C1(A,B;P ). The zero’thgroup is PB with differential d0 : PB → Hom(BAB ,BPB) given by d0(p) =ρp − λp for p ∈ PB.

The second relative Hochschild cochain group is C2(A,B;P ) = Hom (BA⊗BAB ,BPB). The differential at this level is given by d1 : Hom(BAB,BPB) →Hom(BA⊗B AB ,BPB) defined by

(35) (d1f)(x⊗B y) = xf(y)− f(xy) + f(x)y

The cohomology groups are denoted by HHn(A,B;P ) for n ≥ 0; recall or notethat HH1(A,B;P ) is isomorphic to the group of derivations killing B modulothe group of inner derivations w.r.t. elements in PB. For the sake of brevity werefer the reader to textbooks on homological algebra for the details of higherorder cochain groups and differentials.



Note that applications of the hom-tensor relation to the left- and right-handed π-methods with AMA = Hom(AB , PB) yields two isomorphisms

(36) Hom (BA⊗B AB,BPB)∼=−→ Hom(RT ⊗R T,RPBR )

denoted by π2L and π2

R given by

π2L(h)(u ⊗R t) = u1h(u2t1 ⊗B t2)(37)

π2R(h)(u ⊗R t) = h(u1 ⊗B u2t1)t2.(38)

The inverses are given by

π−2L (g)(x⊗B y) =

∑

j,k

γk(xγj(y))g(uk ⊗R uj)(39)

π−2R (g)(x⊗B y) =

∑

i,j

g(ti ⊗R tj)βj(βi(x)y)(40)

Likewise we define the obvious generalized isomorphisms πnL and πnR on then-cochains Cn(A,B;P ).

Brzezinski and Wisbauer define a Hochschild cohomology of an R-coringwith coefficients in an (R,R)-bimodule [1, 30.15]. For the R-coring T and(R,R)-bimodule PB, this specializes to the zero’th cochain group Hom (RTR,

RPBR ) (which is equal to both one-sided R-linear hom-groups considered above

since A |B is irreducible), and first cochain group Hom (RT ⊗R TR,RPBR ). Letε : T → R be the counit of T given by ε(t) = t1t2, the multiplication mappingA⊗BA→ A restricted to T . The differential is given by (h ∈ Hom(RTR,RP

BR ))

(41) (δ0h)(u⊗R t) = ε(u)h(t)− h(u)ε(t)

(42) (δ1g)(u ⊗R v ⊗R t) = ε(u)g(v ⊗R t)− g(uε(v)⊗R t) + g(u⊗R v)ε(t)for g ∈ Hom(RT ⊗R TR,RPBR ). For example, δ1δ0h = 0 by a short computa-tion. The higher cochain groups and differentials are defined similarly and werefer to [1, 30.15] for the details. (This cochain complex recovers Hochschildrelative cochains in case the A-coring A ⊗B A with counit ε′(x ⊗ y) = xytakes the place of R, T and ε.) Denote the cochain groups in this complex byCn(T,R;PB), and call it the Hochschild coring complex.

Next we define a cochain homomorphism Φn : Cn(A,B;P ) → Cn−1(T,R;PB), for n ≥ 1, noting the shift of one downwards in degree in the Hochschildcoring complex. Define Φ1 : Hom (BAB,BPB)→ Hom(RTR,RP

BR ) as

(43) Φ1 = π−1L + π−1

R , Φ1(h)(u) = u1h(u2) + h(u1)u2

for h ∈ Hom(BAB,BPB). Note that Φ1 kills B-linear derivations.Define Φ2 : Hom (BA ⊗B AB,BPB) → Hom(RT ⊗R TR,RP

BR ) by Φ2 =

π2L − π2

R, or in more detail,

(44) (Φ2g)(u⊗R t) = u1g(u2t1 ⊗B t2)− g(u1 ⊗B u2t1)t2

The n’th mapping Φn is easily defined from two obvious generalized mappings,πnL, π

nR : Hom (BA ⊗B · · · ⊗B AB,BPB) → Hom(RT ⊗R · · · ⊗R TR,RPBR ); as

Φn = πnL + (−1)n+1πnR.


196 Lars Kadison

We compute for f ∈ Hom(BAB,BPB),

δ0Φ1f(u⊗R t)= (Φ1f)(u1u2t)− (Φ1f)(ut1t2)

= u1u2t1f(t2) + f(u1u2t1)t2 − f(u1)u2t1t2 − u1f(u2t1t2)

= u1(df)(u2t1 ⊗B t2)− (df)(u1 ⊗B u2t1)t2 = (Φ2d1f)(u⊗R t)

after two middle terms cancel.Moreover, for g ∈ Hom(BA⊗B AB ,BPB),

Φ3(dg)(v ⊗R u⊗R t)= v1(dg)(v2u1 ⊗B u2t1 ⊗B t2) + (dg)(v1 ⊗B v2u1 ⊗B u2t1)t2

= (Φ2g)(ε(v)u⊗R t)− (Φ2g)(vε(u)⊗R t) + (Φ2g)(v ⊗R uε(t))= (δ1Φ2g)(v ⊗R u⊗R t)

after cancellation of the pair of middle terms ±v1g(v2u1 ⊗B u2t1)t2.We omit the tedious but similar computation in degree n which establishes

that Φ is a cochain mapping.

Proposition 5.3. Suppose A |B is an irreducible extension and APA is abimodule. Let R = AB and T = (A ⊗B A)B . Then the mapping of cochaingroups Φn : Cn(A,B;P )→ Cn−1(T,R;PB) is nullhomotopic.

Proof. We define a homotopy sn : Cn+2(A,B;P ) → Cn(T,R : PB) first indegree zero by s0(f)(t) = f(t1 ⊗B t2) where f ∈ Hom(BA ⊗B AB ,BPB),t ∈ T ⊆ A ⊗B A, so f(t) ∈ PB. We claim there is a natural inclusion ιn ofT ⊗R · · · ⊗R T (n times T ) into A⊗B · · · ⊗B A (n+ 1 times A) given by

(45) ιn(u1 ⊗R · · · ⊗R un) = u11 ⊗B u2

1u12 ⊗B · · · ⊗B u2

n−1u1n ⊗B u2

n

In fact, ιn is an isomorphism onto (A⊗B · · ·⊗BA)B which follows from showingthat for any ring C and (A,C)-bimodule M

(46) T ⊗RM∼=−→ A⊗B M

via t⊗Rm 7→ t1 ⊗B t2m. This is a (B,C)-bimodule isomorphism with inversegiven by a⊗B m 7→

∑i ti ⊗R βi(a)m using left D2 quasibases βi ∈ S, ti ∈ T .

Now we may apply this with C = A, M = A ⊗B A,A ⊗B A ⊗B A, . . ., thenrestrict to (−)B, substitute and iterate to prove the claim.

Now define sn(g) = g ιn+1 for g ∈ Cn+2(A,B;P ). We note that δnsn +sn+1d

n+2 = Φn+2; e.g., for f ∈ C2(A,B;P ), u, t ∈ T ,

δ0(s0f)(u⊗R t)+s1(d2f)(u⊗R t) = ε(u)f(t)−f(u)ε(t)+(df)(u1⊗B u2t1⊗B t2)

= u1f(u2t1 ⊗B t2)− f(u1 ⊗B u2t1)t2 = (Φ2f)(u⊗R t)after cancellation of the middle terms in (df)(u1 ⊗B u2t1 ⊗B t2). The generalcase is computed similarly.



6. Appendix: axioms for Hopf algebroids

In this appendix we review the definitions of left bialgebroid, Lu’s Hopfalgebroid, right bialgebroid and the definition of Hopf algebroid by Bohm-Szlachanyi. First, for the definition of a left bialgebroid (H,R, sL, tL,∆, ε), Hand R are K-algebras and all maps are K-linear. First, recall from [16] thatthe source and target maps sL and tL are algebra homomorphism and anti-homomorphism, respectively, of R into H such that sL(r)tL(s) = tL(s)sL(r)for all r, s ∈ R. This induces an R-R-bimodule structure on H (from the leftin this case) by r · h · s = sL(r)tL(s)h (h ∈ H). With respect to this bimodulestructure, (H,∆, ε) is an R-coring (cp. [18]), i.e. with coassociative coproductand R-R-bimodule map ∆ : H → H ⊗R H and counit ε : H → R (also an R-bimodule mapping). The image of ∆, written in Sweedler notation, is requiredto satisfy

(47) a(1)tL(r)⊗ a(2) = a(1) ⊗ a(2)sL(r)

for all a ∈ H, r ∈ R. It then makes sense to require that ∆ be homomorphic:

(48) ∆(ab) = ∆(a)∆(b), ∆(1) = 1⊗ 1

for all a, b ∈ H . The counit must satisfy the following modified augmentationlaw:

(49) ε(ab) = ε(as(ε(b))) = ε(at(ε(b))), ε(1H) = 1R.

The axioms of a right bialgebroid H ′ are opposite those of a left bialgebroidin the sense that H ′ obtains its R-bimodule structure from the right via itssource and target maps and, from the left bialgebroid H above, we have that(Hop, R, topL , s

opL ,∆, ε) (in that precise order) is a right bialgebroid: for the

explicit axioms, see [9, Sec. 2].In addition, the left R-bialgebroid H is a Hopf algebroid in the sense of Lu

(H,R, τ) if (antipode) τ : H → H is an algebra anti-automorphism such that

(1) τtL = sL;(2) τ(a(1))a(2) = tL(ε(τ(a))) for every a ∈ A;(3) there is a linear section η : H⊗RH → H⊗KH to the natural projection

H ⊗K H → H ⊗R H such that:

µ(H ⊗ τ)η∆ = sLε.

The following is one of several equivalent definitions of Bohm-Szlachanyi’sHopf algebroid [2], excerpted from [9, 8.7].

Definition 6.1. We call H a Hopf algebroid if there are left and right bialge-broid structures (H,R, sL, tL,∆L, εL) and (H,Rop, sR, tR,∆R, εR) such that

(1) Im sR = Im tL and Im tR = Im sL,(2) (1⊗∆L)∆R = (∆R ⊗ 1)∆L and (1⊗∆R)∆L = (∆L ⊗ 1)∆R

with anti-automorphism τ : H → H (called an antipode) such that

(i): τ(atR(r)) = sR(r)τ(a) and τ(tL(r)a) = τ(a)sL(r) for r ∈ R, a ∈ H ,(ii): a(1)τ(a(2)) = sL(εL(a)) and τ(a(1))a(2) = sR(εR(a))


198 Lars Kadison

where ∆R(a) = a(1) ⊗ a(2) and ∆L(a) = a(1) ⊗ a(2).

The relationship between Lu’s Hopf algebroid and this alternative Hopfalgebroid with more pleasant tensor categorical properties is discussed in [2, 3]and other papers by Bohm and Szlachanyi.

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[2] G. Bohm and K. Szlachanyi, Hopf algebroids with bijective antipodes: axioms, integrals,and duals, J. Algebra 274 (2004), no. 2, 708–750. MR2043373 (2004m:16047)

[3] G. Bohm, Integral theory for Hopf algebroids, Algebr. Represent. Theory 8 (2005),no. 4, 563–599. MR2199210 (2006m:16048)

[4] J. Cuadra, A Hopf algebra having a separable Galois extension is finite dimensional,Proc. Amer. Math. Soc. 136 (2008), no. 10, 3405–3408. MR2415022 (2009c:16116)

[5] Y. Doi and M. Takeuchi, Hopf-Galois extensions of algebras, the Miyashita-Ulbrichaction, and Azumaya algebras, J. Algebra 121 (1989), no. 2, 488–516. MR0992778(90b:16015)

[6] M. Grana, J. A. Guccione and J. J. Guccione, Decomposition of some pointed Hopfalgebras given by the canonical Nakayama automorphism, J. Pure Appl. Algebra 210

(2007), no. 2, 493–500. MR2320012 (2009d:16058)[7] L. Kadison, New examples of Frobenius extensions, Amer. Math. Soc., Providence, RI,

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depth two, Adv. Math. 163 (2001), no. 2, 258–286. MR1864835 (2003h:46098)[9] L. Kadison and K. Szlachanyi, Bialgebroid actions on depth two extensions and duality,

Adv. Math. 179 (2003), no. 1, 75–121. MR2004729 (2004i:16055)[10] L. Kadison, Hopf algebroids and H-separable extensions, Proc. Amer. Math. Soc. 131

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Received November 5, 2008; accepted January 7, 2009

Lars KadisonDepartment of Mathematics, University of PennsylvaniaDavid Rittenhouse Lab, 209 S. 33rd St., Philadelphia, PA 19104currently: Louisiana State University, Baton Rouge, LA 70803E-mail: [email protected]

URL: www.math.upenn.edu/~lkadison




On the classifying space of the family of

virtually cyclic subgroups or CAT(0)-groups

Wolfgang Luck



Abstract. Let G be a discrete group which acts properly and isometrically on a completeCAT(0)-space X. Consider an integer d with d = 1 or d ≥ 3 such that the topologicaldimension of X is bounded by d. We show the existence of a G-CW -model EG for theclassifying space for proper G-actions with dim(EG) ≤ d. Provided that the action is alsococompact, we prove the existence of a G-CW -model EG for the classifying space of the

family of virtually cyclic subgroups satisfying dim(EG) ≤ d + 1.

1. Introduction

Given a group G, denote by EG a G-CW -model for the classifying space forproper G-actions and by EG = EVCY(G) a G-CW -model for the classifyingspace of the family of virtually cyclic subgroups. Our main theorem which willbe proved in Section 4 is

Theorem 1.1. Let G be a discrete group which acts properly and isometri-cally on a complete proper CAT(0)-space X. Let top-dim(X) be the topologicaldimension of X. Let d be an integer satisfying d = 1 or d ≥ 3 such thattop-dim(X) ≤ d.

(i) Then there is G-CW -model EG with dim(EG) ≤ d;(ii) Suppose that G acts by semisimple isometries. (This is the case if we

additionally assume that the G-action is cocompact.)Then there is G-CW -model EG with dim(EG) ≤ d+ 1.

The paper was supported by the Sonderforschungsbereich 478—Geometrische Struk-turen in der Mathematik—and the Max-Planck-Forschungspreis and the Leibniz-Preis of

the author.

202 Wolfgang Luck

There is the question whether for any group G the inequality

(1.2) hdimG(EG)− 1 ≤ hdimG(EG) ≤ hdimG(EG) + 1

holds, where hdimG(EG) is the minimum of the dimensions of all possible

G-CW -models for EG and hdimG(EG) is defined analogously (see [15, Intro-

duction]). Since hdim(EG) ≤ 1 + hdim(EG) holds for all groups G (see [15,Corollary 5.4]), Theorem 1.1 implies

Corollary 1.3. Let G be a discrete group and let X be complete CAT(0)-spaceX with finite topological dimension top-dim(X). Suppose that G acts properlyand isometrically on X. Assume that the G-action is by semisimple isometries.(The last condition is automatically satisfied if we additionally assume that the

G-action is cocompact.) Suppose that top-dim(X) = hdimG(EG) 6= 2.Then inequality (1.2) is true.

We will prove at the end of Section 4

Corollary 1.4. Suppose that G is virtually torsionfree. Let M be a simplyconnected complete Riemannian manifold of dimension n with non-negativesectional curvature. Suppose that G acts on M properly, isometrically andcocompactly. Then

hdim(EG) = n;n− 1 ≤ hdim(EG) ≤ n+ 1.

In particular (1.2) holds.

If G is the fundamental group of an n-dimensional closed hyperbolic man-ifold, then hdim(EG) = hdim(EG) = n by [15, Example 5.12]. If G is vir-

tually Zn for n ≥ 2, then hdim(EG) = n and hdim(EG) = n + 1 by [15,

Example 5.21]. Hence the cases hdim(EG) = hdim(EG) and hdim(EG) =

hdim(EG) + 1 do occur in the situation of Corollary 1.4. There exists groupsG with hdim(EG) = hdim(EG) − 1 (see [15, Example 5.29]). But we do notbelieve that this is possible in the situation of Corollary 1.3 or Corollary 1.4.

In the preprint by Farley [9] constructions for EG are given for a group Gacting by semisimple isometries on a proper CAT(0)-space under the assump-tion that there are some G-well-behaved spaces of axes.

The author wants to thank the referee for his valuable suggestions.

2. Classifying Spaces for Families

We briefly recall the notions of a family of subgroups and the associatedclassifying space. For more information, we refer for instance to the originalsource [18] or to the survey article [13].


On the classifying space 203

A family F of subgroups of G is a set of subgroups of G which is closedunder conjugation and taking subgroups. Examples for F are

1 = trivial subgroup;FIN = finite subgroups;VCY = virtually cyclic subgroups;ALL = all subgroups.

Let F be a family of subgroups of G. A model for the classifying spaceEF (G) of the family F is a G-CW -complex X all of whose isotropy groupsbelong to F such that for any G-CW -complex Y with isotropy groups in Fthere exists a G-map Y → X and any two G-maps Y → X are G-homotopic.In other words, X is a terminal object in the G-homotopy category of G-CW -complexes whose isotropy groups belong to F . In particular, two models forEF (G) are G-homotopy equivalent.

There exists a model for EF (G) for any group G and any family F of sub-groups. There is even a functorial construction (see [6, page 223 and Lemma 7.6(ii)]).

A G-CW -complex X is a model for EF (G) if and only if the H-fixed pointset XH is contractible for H ∈ F and is empty for H 6∈ F .

We abbreviate EG := EFIN (G) and call it the universal G-CW -complexfor proper G-actions. We also abbreviate EG := EVCY(G).

A model for EALL(G) is G/G. A model for E1(G) is the same as a modelfor EG, which denotes the total space of the universal G-principal bundleEG→ BG.

One can also define a numerable version of the space for proper G-actionsto G which is denoted by JG. It is not necessarily a G-CW -complex. Ametric space X on which G acts isometrically and properly is a model for JGif and only if the two projections X ×X → X onto the first and second factorare G-homotopic to one another. If X is a complete CAT(0)-space on whichG-acts properly and isometrically, then X is a model for JG, the desired G-homotopy is constructed using the geodesics joining two points in X (see [4,Proposition 1.4 in II.1 on page 160]).

One motivation for studying the spaces EG and EG comes from the Baum-Connes Conjecture and the Farrell-Jones Conjecture. For more informationabout these conjectures we refer for instance to [2, 10, 14, 16].

3. Topological and CW -dimension

Let X be a topological space. Let U be an open covering. Its dimensiondim(U) ∈ 0, 1, 2, . . .∐∞ is the infimum over all integers d ≥ 0 such that forany collection U0, U1, . . . , Ud of pairwise distinct elements in U the intersection⋂di=0 Ui is empty. An open covering V is a refinement of U if for every V ∈ V

there is U ∈ U with V ⊆ U .


204 Wolfgang Luck

Definition 3.1 (Topological dimension). The topological dimension (some-times also called covering dimension) of a topological space X

top-dim(X) ∈ 0, 1, 2, . . . ∐ ∞

is the infimum over all integers d ≥ 0 such that any open covering U possessesa refinement V with dim(V) ≤ d.

Let Z be a metric space. We will denote for z ∈ Z and r ≥ 0 by Br(z)and Br(z) respectively the open ball and closed ball respectively around z withradius r. We call Z proper if for each z ∈ Z and r ≥ 0 the closed ball Br(z) iscompact. A group G acts properly on the topological space Z if for any z ∈ Zthere is an open neighborhood U such that the set g ∈ G | g · U ∩ U 6= ∅ isfinite. In particular every isotropy group is finite. If Z is a G-CW -complex,then Z is a proper G-space if and only if the isotropy group of any point in Zis finite (see [12, Theorem 1.23]).

Lemma 3.2. Let Z be a proper metric space. Suppose that G acts on Zisometrically and properly. Then we get for the topological dimensions of Xand G\X

top-dim(G\X) ≤ top-dim(X).

Proof. Since G acts properly and isometrically, we can find for every z ∈ Z areal number ǫ(z) > 0 such that we have for all g ∈ G

g ·B7ǫ(z)(z) ∩B7ǫ(z) 6= ∅ ⇐⇒ g ·B7ǫ(z)(z) = B7ǫ(z)(z) ⇐⇒ g ∈ Gz.

We can arrange that ǫ(gz) = ǫ(z) holds for z ∈ Z and g ∈ G. ConsiderG · Bǫ(z). We claim that this set is closed in Z. We have to show for asequence (zn)n≥0 of elements in Bǫ(z) and (gn)n≥0 of elements in G and x ∈ Zwith limn→∞ gnzn = x that x belongs to G · Bǫ(z). Since X is proper, wecan find y ∈ Bǫ(z) such that limn→∞ zn = y. Choose N = N(ǫ) such thatdX(gnzn, x) ≤ ǫ and dX(zn, y) ≤ ǫ holds for n ≥ N . We conclude for n ≥ N

dx(gny, x) ≤ dX(gny, gnzn) + dX(gnzn, x)

= dX(y, zn) + dX(gnzn, x)

≤ ǫ+ ǫ

= 2ǫ.

This implies for n ≥ N

dX(g−1n gNz, z) = dX(gNz, gnz)

≤ dX(gNz, gNy) + dX(gNy, x) + dX(x, gny) + dX(gny, gnz)

= dX(z, y) + dX(gNy, x) + dX(gny, x) + dX(y, z)

≤ ǫ+ 2ǫ+ 2ǫ+ ǫ

= 6ǫ.



Hence g−1n gN ∈ Gz for n ≥ N . Since Gz is finite, we can arrange by passing

to subsequences that g0 = gn holds for n ≥ 0. Hence

x = limn→∞

gnzn = limn→∞

g0zn = g0 · limn→∞

zn = g0 · y ∈ G ·Bǫ(z).

Choose a set-theoretic section s : G/Gz → G of the projection G → G/Gz.The map

G/Gz ×B7ǫ(z)(z)∼=−→ G · B7ǫ(z)(z), (gGz , x) 7→ s(gGz) · x

is bijective, continuous and open and hence a homeomorphism. It induces ahomeomorphism

G/Gz ×Bǫ(z)(z)∼=−→ G · Bǫ(z)(z).

This implies

(3.3) top-dim(Bǫ(z)(z)

)= top-dim

(G · Bǫ(z)(z)

).

Let pr : Z → G\Z be the projection. It induces a bijective continuous map

Gz\Bǫ(z)(z)∼=−→ pr

(Bǫ(z)(z)

)which is a homeomorphism since Bǫ(z)(z) and

hence Gz\Bǫ(z)(z) is compact. Hence we get

(3.4) top-dim(pr(Bǫ(z)(z))

)= top-dim

(Gz\Bǫ(z)(z)

).

Since the metric space Bǫ(z)(z) is compact and hence contains a countabledense set and Gz is finite, we conclude from [3, Exercise in Chapter II onpage 112]

(3.5) top-dim(Gz\Bǫ(z)(z)

)≤ top-dim

(Bǫ(z)(z)

).

From (3.3), (3.4) and (3.5) we conclude thatG·Bǫ(z)(z) ⊆ Z and pr(Bǫ(z)(z)

)⊆

G\Z are closed and satisfy

(3.6) top-dim(pr(Bǫ(z)(z))

)≤ top-dim

(G ·Bǫ(z)(z)

).

Since Z is proper, it is the countable union of compact subspaces and hencecontains a countable dense subset. This is equivalent to the condition that Zhas a countable basis for its topology. Obviously the same is true for G\Z. Weconclude from [17, Theorem 9.1 in in Chapter 7.9 on page 302 and Exercise 9in Chapter 7.9 on page 315]

top-dim(Z) = suptop-dim

(G ·Bǫ(z)(z)

);(3.7)

top-dim(G\Z) = suptop-dim

(pr(Bǫ(z)(z))

).(3.8)

Now Lemma 3.2 follows from (3.6), (3.7) and (3.8).

In the sequel we will equip a simplicial complex with the weak topology,i.e., a subset is closed if and only if its intersection with any simplex σ is aclosed subset of σ. With this topology a simplicial complex carries a canonicalCW -structure.

Let X be a G-space. We call a subset U ⊆ X a FIN -set if we havegU ∩ U 6= ∅ =⇒ gU = U for every g ∈ G and GU := g ∈ G | g · U = Uis finite. Let U be a covering of X by open FIN -subset. Suppose that U is


206 Wolfgang Luck

G-invariant, i.e., we have g · U ∈ U for g ∈ G and U ∈ U . Define its nerveN (U) to be the simplicial complex whose vertices are the elements in U andfor which the pairwise distinct vertices U0, U1, . . . , Ud span a d-simplex if and

only if⋂di=0 Ui 6= ∅. The action of G on X induces an action on U and hence

a simplicial action on N (U). The isotropy group of any vertex is finite andhence the isotropy group of any simplex is finite. Let N (U)′ be the barycentricsubdivision. It inherits a simplicial G-action from N (U) such that for anyg ∈ G and any simplex σ whose interior is denoted by σ and which satisfiesg · σ ∩ σ 6= ∅ we have gx = x for all x ∈ σ. In particular N (U)′ is aG-CW -complex and agrees as a G-space with N (U).

Lemma 3.9. Let n be an integer with n ≥ 0. Let X be a proper metric spacewhose topological dimension satisfies top-dim(X) ≤ n. Suppose that G actsproperly and isometrically on X.

Then there exists a proper n-dimensional G-CW -complex Y together with aG-map f : X → Y .

Proof. Since the G-action is proper we can find for every x ∈ X an ǫ(x) > 0such that for every g ∈ G we have

g · B2ǫ(x)(x) ∩B2ǫ(x)(x) 6= ∅ ⇔ g ·B2ǫ(x)(x) = B2ǫ(x)(x)

⇔ g ·B2ǫ(x)(x) = B2ǫ(x)(x) ⇔ g ·Bǫ(x)(x) = Bǫ(x)(x) ⇔ g ∈ Gx.We can arrange that ǫ(gx) = ǫ(x) for g ∈ G and x ∈ X holds. We obtaina covering of X by open FIN -subsets

Bǫ(x)(x) | x ∈ X

. Let pr : X →

G\X be the canonical projection. We obtain an open covering of G\X bypr(Bǫ(x)(x)

)| x ∈ X

. Since top-dim(X) ≤ n by assumption and G acts

properly on X , we get top-dim(G\X) ≤ n from Lemma 3.2. Since G actsproperly and isometrically on X , the quotient G\X inherits a metric fromX . Hence G\X is paracompact by Stone’s theorem (see [17, Theorem 4.3in Chap. 6.3 on page 256]) and in particular normal. By [7, Theorem 3.5on page 211] we can find a locally finite open covering U of G\X such thatdim(U) ≤ n and U is a refinement of

pr(Bǫ(x)(x)) | x ∈ X

. For each U ∈ U

choose x(U) ∈ X with U ⊆ pr(Bǫ(U)(x(U)

). Define the index set

J =(U, g) | U ∈ U , g ∈ G/Gx(U)

.

For (U, g) ∈ J define an open FIN -subset of X by

VU,g := pr−1(U) ∩ g ·B2ǫ(x(U))

(x(U)

).

Obviously this is well-defined, i.e., the choice of g ∈ g does not matter, and wehave pr(VU,g) ⊆ U and VU,g ⊆ g · B2ǫ(x(U))

(x(U)

).

Consider the collection of subsets of X

V =VU,g | (U, g) ∈ J.

This is a G-invariant covering of X by open FIN -subsets. Its dimensionsatisfies

dim(V) ≤ dim(U) ≤ n



since for U ∈ U , g1, g2 ∈ G/Gx(U) we have

VU,g1 ∩VU,g2 6= ∅ =⇒ g1 ·B2ǫ(x(U))

(x(U)

)∩g2 ·B2ǫ(x(U))

(x(U)

)=⇒ g1 = g2.

Since U is locally finite and G\X is paracompact, we can find a locally finitepartition of unity

eU : G\X → [0, 1] | U ∈ U

which is subordinate to U , i.e.,∑

U∈U eU = 1 and supp(eU ) ⊂ U for every U ∈ U . Fix a map χ : [0,∞)→ [0, 1]

satisfying χ−1(0) = [1,∞). Define for (U, g) ∈ J a function

φU,g : X → [0, 1], y 7→ eU (pr(y)) · χ(dX(y, gx(U))/ǫ(x(U))

).

Consider y ∈ X . Since U is locally finite and G\X is locally compact, wecan find an open neighborhood T of pr(y) such that T meets only finitely manyelements of U . Choose an open neighborhoodW0 of y such that W0 is compact.Define an open neighborhood of y by

W := W0 ∩ pr−1(T ).

Since W0 is compact, W is compact. Since G acts properly, there exists for agiven U ∈ U only finitely many elements g ∈ G with W∩g ·Bǫ(x(U))(x(U)) 6= ∅.

Since T meets only finitely elements of U , the set

JW :=(U, g) ∈ J |W ∩ g ·Bǫ(x(U))(x(U)) ∩ pr−1(U) 6= ∅

is finite. Suppose φU,g(z) > 0 for (U, g) ∈ J and z ∈ W . We concludez ∈ pr−1(U) ∩ g · Bǫ(x(U))(x(U)) and hence (U, g) ∈ JW . Thus we have shown

that the collectionφU,g | (U, g) ∈ J

is locally finite.

We conclude that the map∑

(U,g)∈J

φU,g : X → [0, 1], y 7→∑

(U,g)∈J

eU (pr(y)) · χ(dX(y, gx(U))/ǫ(x(U))

)

is well-defined and continuous. It has always a value greater than zero sincefor every y ∈ X there exists U ∈ U with eU (pr(y)) > 0, the set pr−1(U) iscontained in

⋃g∈G g · Bǫ(U)(x(U)) and χ−1(0) = [1,∞). Define for (U, g) ∈ J

a map

ψU,g : X → [0, 1], y 7→ φU,g(y)∑(U,g)∈J φU,g(y)

.

We conclude that∑(U,g)∈J ψU,g(y) = 1 for y ∈ X ;

ψU,g(hy) = ψU,h−1g

(y) for h ∈ G, y ∈ Y and (U, g) ∈ J ;

supp(ψU,g) ⊆ VU,g for (U, g) ∈ J,and the collection

ψU,g | (U, g) ∈ J

is locally finite. Define the desired

proper n-dimensional G-CW -complex to be the nerve Y := N (V). Define amap by

f : X → N (V), y 7→∑

(U,g)∈J

ψU,g(y) · VU,g.

It is well-defined since for y ∈ X the simplices VU,g for which ψU,g(y) 6= 0 holdsspan a simplex because y ∈ X with ψU,g(y) 6= 0 belongs to VU,g and hence the


208 Wolfgang Luck

intersection of the sets VU,g for which ψU,g(y) 6= 0 holds contains y and hence isnonempty. The map f is continuous since

ψU,g | (U, g) ∈ J

is locally finite.

It is G-equivariant by the following calculation for h ∈ G and y ∈ Y :

f(hy) =∑

(U,g)∈J

ψU,g(hy) · VU,g

=∑

(U,g)∈J

ψU,hg(hy) · VU,hg

=∑

(U,g)∈J

ψU,h−1hg

(y) · VU,hg

=∑

(U,g)∈J

ψU,g(y) · h · VU,g

= h ·∑

(U,g)∈J

ψU,g(y) · VU,g

= h · f(y).

Lemma 3.10. Let X and Y be G-CW -complexes. Let i : X → Y and r : Y →X be G-maps such that ri is G-homotopic to the identity map on X. Consideran integer d ≥ 3. Suppose that Y has dimension ≤ d.

Then X is G-homotopy equivalent to a G-CW -complex Z of dimension ≤ d.Proof. By the Equivariant Cellular Approximation Theorem (see [19, Theo-rem II.2.1 on page 104]) we can assume without loss of generality that i andr are cellular. Let cyl(r) be the mapping cylinder. Let k : Y → cyl(r) be thecanonical inclusion and p : cyl(r) → X be the canonical projection. Then pis a G-homotopy equivalence and p k = r. Let Z be the union of the 2-skeleton of cyl(r) and Y . This is a G-CW -subcomplex of cyl(r) and cyl(r) isobtained from Z by attaching equivariant cells of dimension ≥ 3. Hence themap p|Z : Z → X has the property that it induces on every fixed point seta 2-connected map. Let j : X → Z be the composite of i : X → Y with theobvious inclusion Y → Z. Then p|Z j = p k i = r i is G-homotopyequivalent to the identity and the dimension of Z is still bounded by d sincewe assume d ≥ 3. Hence we can assume in the sequel that rH : Y H → XH is2-connected for all H ⊆ G, otherwise replace Y by Z, i by j and r by p|Z .

We want to apply [12, Proposition 14.9 on page 282]. (We will use thenotation of this reference that for a category C a ZC-module or a ZC-chaincomplex respectively is a contravariant functor from C to the category of Z-modules or of Z-chain complexes respectively.) Here the assumption d ≥ 3enters. Hence it suffices to show that the cellular ZΠ(G,X)-chain complexCc∗(X) is ZΠ(G,X)-chain homotopy equivalent to a d-dimensional ZΠ(G,X)-chain complex. By [12, Proposition 11.10 on page 221] it suffices to show thatthe cellular ZΠ(G,X)-chain complex Cc∗(X) is dominated by a d-dimensionalZΠ(G,X)-chain complex. This follows from the geometric domination (Y, i, r)



by passing to the cellular chain complexes over the fundamental categories sincer and hence also i induce equivalences between the fundamental categoriesbecause rH : Y H → XH is 2-connected for all H ⊆ G and r i ≃G idX .

The condition d ≥ 3 is needed since we want to argue first with the cellularZOr(G)-chain complex and then transfer the statement that it is d-dimensionalto the statement that the underlying G-CW -complex is d-dimensional. Thecondition d ≥ 3 enters for analogous reasons in the classical proof of the the-orem that the existence of a d-dimensional ZG-projective resolution for thetrivial ZG-module Z implies the existence of a d-dimensional model for BG(see [5, Theorem 7.1 in Chapter VIII.7 on page 205]).

Theorem 3.11. Let G be a discrete group. Then

(i) There is a G-homotopy equivalence JG→ EG;(ii) Suppose that there is a model for JG which is a metric space such that

the action of G on JG is isometric. Consider an integer d with d = 1or d ≥ 3. Suppose that the topological dimension top-dim(JG) ≤ d.

Then there is a G-CW -model for EG of dimension ≤ d;(iii) Let d be an integer d ≥ 0. Suppose that there is a G-CW -model for

EG with dim(EG) ≤ d such that EG after forgetting the group actionhas countably many cells.

Then there exists a model for JG with top-dim(JG) ≤ d.Proof. (i) This is proved in [13, Lemma 3.3 on page 278].

(ii) Choose a G-homotopy equivalence i : EG → JG. From Lemma 3.9 weobtain a G-map f : JG → Y to a proper G-CW -complex of dimension ≤ d.By the universal properly of EG we can find a G-map h : Y → EG and thecomposite h f i is G-homotopic to the identity on EG.

Suppose d ≥ 3. We conclude from Lemma 3.10 that EG is G-homotopyequivalent to a G-CW -complex of dimension ≤ d.

Suppose d = 1. By Dunwoody [8, Theorem 1.1] it suffices to show that therational cohomological dimension of G satisfies cdQ(G) ≤ 1. Hence we haveto show for any QG-module M that ExtnQG(Q,M

)= 0 for n ≥ 2, where Q

is the trivial QG-module. Since all isotropy groups of EG and Y are finite,their cellular QG-chain complexes are projective. Since EG is contractible,C∗(EG; Q) is a projective QG-resolution and hence

ExtnQG(Q,M) ∼= Hn

(homQG(C∗(EG; Q),M)

).

Since h f i ≃G idEG, the Q-module Hn(homQG(C∗(EG; Q),M)

)is a di-

rect summand in the Q-module Hn(homQG(C∗(Y ; Q),M)

). Since Y is 1-

dimensional by assumption, Hn(homQG(C∗(Y ; Q),M)

)vanishes for n ≥ 2.

This implies that ExtnQG(Q,M)

vanishes for n ≥ 2.

(iii) Using the equivariant version of the simplicial approximation theorem andthe fact that changing the G-homotopy class of attaching maps does not change


210 Wolfgang Luck

the G-homotopy type, one can find a simplicial complex X with simplicial G-action which is G-homotopy equivalent to EG, satisfies dim(X) = dim(EG)and has only countably many simplices. Hence the barycentric subdivisionX ′ is a simplicial complex of dimension ≤ d with countably many simplicesand carries a G-CW -structure. The latter implies that X ′ is a G-CW -modelfor EG and hence also a model for JG. Since the dimension of a simplicialcomplex with countably many simplices is equal to its topological dimension,we conclude top-dim(X ′) = dim(X) = dim(EG) ≤ d.

Remark 3.12. The referee has pointed out to the author that one can givea simplified and improved version of assertion (iii) of Theorem 3.11. Namely,one can replace the hypothesis just by the hypothesis that G is countable.

If there is a G-CW -model for EG such that EG after forgetting the groupaction has countably many 0-cells, then G is countable.

By inspecting the proof one realizes that the condition that G is countablesuffices to conclude the existence of a model for JG with top-dim(JG) ≤ dwhich has only countably many cells after forgetting the group action.

4. The passage from finite to virtually cyclic groups

In [15] it is described how one can construct EG from EG. In this sectionwe want to make this description more explicit under the following condition

Condition 4.1. We say that G satisfies condition (C) if for every g, h ∈ Gwith |h| =∞ and k, l ∈ Z we have

ghkg−1 = hl =⇒ |k| = |l|.Let ICY be the set of infinite cyclic subgroup C of G. This is not a family

since it does not contain the trivial subgroup. We call C,D ∈ ICY equivalentif |C ∩D| =∞. One easily checks that this is an equivalence relation on ICY .Denote by [ICY] the set of equivalence classes and for C ∈ ICY by [C] itsequivalence class. Denote by

NGC := g ∈ G | gCg−1 = Cthe normalizer of C in G. Define for [C] ∈ [ICY] a subgroup of G by

NG[C] :=g ∈ G

∣∣ |gCg−1 ∩ C| =∞.

This is the same the commensurator of the subgroup C ⊆ G, i.e., the set ofelements g ∈ G for which H ∩ gHg−1 has finite index in both H and gHg−1.One easily checks that this is independent of the choice of C ∈ [C]. ActuallyNG[C] is the isotropy of [C] under the action of G induced on [ICY] by theconjugation action of G on ICY.

Lemma 4.2. Suppose that G satisfies Condition (C) (see 4.1). Consider C ∈ICY.

Then obtain a nested sequence of subgroups

NGC ⊆ NG2!C ⊆ NG3!C ⊆ NG4!C ⊆ · · ·



where k!C is the subgroup of C given by hk! | h ∈ C, and we have

NG[C] =⋃

k≥1

NGk!C.

Proof. Since every subgroup of a cyclic group is characteristic, we obtain thenested sequence of normalizers NGC ⊆ NG2!C ⊆ NG3!C ⊆ NG4!C ⊆ · · · .

Consider g ∈ NG[C]. Let h be a generator of C. Then there are k, l ∈ Zwith ghkg−1 = hl and k, l 6= 0. Condition (C) implies k = ±l. Hence g ∈NG〈hk〉 ⊆ NGk!C. This implies NG[C] ⊆ ⋃k≥1NGk!C. The other inclusion

follows from the fact that for g ∈ NGk!C we have k!C ⊆ gCg−1 ∩C.

Fix C ∈ ICY . Define a family of subgroups of NG[C] by

(4.3) GG(C) :=H ⊆ NG[C] | [H : (H ∩ C)] <∞

∪H ⊆ NG[C] | |H | <∞

.

Notice that GG(C) consists of all finite subgroups of NG[C] and of all virtuallycyclic subgroups of NG[C] which have an infinite intersection with C. Definea quotient group of NGC by

WGC := NGC/C.

Lemma 4.4. Let n be an integer. Suppose that G satisfies Condition (C)(see 4.1). Suppose that there exists a G-CW -model for EG with dim(EG) ≤n and for every C ∈ ICY there exists a WGC-CW -model for EWGC withdim(EWGC) ≤ n.

Then there exists a G-CW -model for EG with dim(EG) ≤ n+ 1.

Proof. Because of [15, Theorem 2.3 and Remark 2.5] it suffices to show forevery C ∈ ICY that there is a NG[C]-model for EGG(C)(NG[C]) with

(4.5) dim(EGG(C)(NG[C])) ≤ n+ 1.

Because of Lemma 4.2 we have

NG[C] = colimk→∞NGk!C.

We conclude (4.5) from [15, Lemma 4.2 and Theorem 4.3] since every elementH ∈ GG(C) is finitely generated and hence lies already in NGk!C for somek > 0, by assumption there exists a WGk!C-CW -model for EWGk!C withdim(EWGk!C) ≤ n, and resNGk!C→WGk!C EWGk!C is EGG(C)|NGk!C)

(NGk!C).

Now we are ready to prove Theorem 1.1.Proof of Theorem 1.1. (i) Consider an integer d ∈ Z with d = 1 or d ≥ 3 suchthat d ≥ top-dim(X). The space X is a model for JG by [4, Corollary 2.8in II.2. on page 178]. We conclude from Theorem 3.11 (ii) that there is ad-dimensional model for EG.

(ii) We will use in the proof some basic facts and notions about isometries ofproper complete CAT(0)-spaces which can be found in [4, Chapter II.6].


212 Wolfgang Luck

The group G satisfies condition (C) by the following argument. Supposethat ghkg−1 = hl for g, h ∈ G with |h| = ∞ and k, l ∈ Z. The isometrylh : X → X given by multiplication with h is a hyperbolic isometry since it hasno fixed point and is by assumption semisimple. We obtain for the translationlength L(h) which is a real number satisfying L(h) > 0

|k| · L(h) = L(hk) = L(ghkg−1) = L(hl) = |l| · L(h).

This implies |k| = |l|.Let C ⊆ G be any infinite cyclic subgroup. Choose a generator g ∈ C. The

isometry lg : X → X given by multiplication with g is a hyperbolic isometry.Let Min(g) ⊂ X be the the union of all axes of g. Then Min(g) is a closedconvex subset of X . There exists a closed convex subset Y (g) ⊆ X and anisometry

α : Min(g)∼=−→ Y (g)× R.

The space Min(G) isNGC-invariant since for each h ∈ NGC we have hgh−1 = gor hgh−1 = g−1 and hence multiplication with h sends an axis of g to an axisof g. The NGC-action induces a proper isometric WGC-action on Y (g). Theseclaims follow from [4, Theorem 6.8 in II.6 on page 231 and Proposition 6.10in II.6 on page 233]. The space Y (g) inherits fromX the structure of a CAT(0)-space and satisfies top-dim(Y (g)) ≤ top-dim(X). Hence Y (g) is a model forJWGC with top-dim(Y (g)) ≤ top-dim(X) by [4, Corollary 2.8 in II.2. onpage 178]. We conclude from Theorem 3.11 (ii) that there is a d-dimensionalmodel for EWGC for every infinite cyclic subgroup C ⊆ G. Now Theorem 1.1follows from Lemma 4.4.

Finally we prove Corollary 1.4.

Proof of Corollary 1.4. A complete Riemannian manifoldM with non-negativesectional curvature is a CAT(0)-space (see [4, Theorem IA.6 on page 173 andTheorem II.4.1 on page 193].) Since G is virtually torsionfree, we can find asubgroupG0 of finite index in G such that G0 is torsionfree and acts orientationpreserving on M . Hence G0\M is a closed orientable manifold of dimension n.Hence Hn(M ; Z) = Hn(BG; Z) 6= 0. This implies that every CW -model BG0

has at least dimension n. Since the restriction of EG to G0 is a G0-CW -modelfor EG0, we conclude hdim(EG) ≥ n. Since M with the given G0-action is aG-CW -model for EG (see [1, Theorem 4.15]), we conclude

hdim(EG) = n = top-dim(M).

If n 6= 2, we conclude hdim(EG) ≤ n+1 from Theorem 1.1. Since hdim(EG) ≤1 + hdim(EG) holds for all groups G (see [15, Corollary 5.4]), we get

n− 1 ≤ hdim(EG) ≤ n+ 1

provided that n 6= 2.Suppose n = 2. If G0 is a torsionfree subgroup of finite index in G, then

G0\X is a closed 2-dimensional manifold with non-negative sectional curvature.Hence G0 is Z2 or hyperbolic. This implies that G is virtually Z2 or hyperbolic.



Hence hdim(EG) ∈ 2, 3 by [15, Example 5.21] in the first case and by [15,

Theorem 3.1, Example 3.6, Theorem 5.8 (ii)] or [11, Proposition 6, Remark 7and Proposition 8] in the second case.

References

[1] H. Abels, A universal proper G-space, Math. Z. 159 (1978), no. 2, 143–158. MR0501039(58 #18504)

[2] P. Baum, A. Connes and N. Higson, Classifying space for proper actions and K-theoryof group C∗-algebras, in C∗-algebras: 1943–1993 (San Antonio, TX, 1993), 240–291,Contemp. Math., 167, Amer. Math. Soc., Providence, RI. MR1292018 (96c:46070)

[3] G. E. Bredon, Introduction to compact transformation groups, Academic Press, NewYork, 1972. MR0413144 (54 #1265)

[4] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer,Berlin, 1999. MR1744486 (2000k:53038)

[5] K. S. Brown, Cohomology of groups, Springer, New York, 1982. MR0672956 (83k:20002)[6] J. F. Davis and W. Luck, Spaces over a category and assembly maps in isomorphism

conjectures in K- and L-theory, K-Theory 15 (1998), no. 3, 201–252. MR1659969(99m:55004)

[7] C. H. Dowker, Mapping theorems for non-compact spaces, Amer. J. Math. 69 (1947),200–242. MR0020771 (8,594g)

[8] M. J. Dunwoody, Accessibility and groups of cohomological dimension one, Proc. Lon-don Math. Soc. (3) 38 (1979), no. 2, 193–215. MR0531159 (80i:20024)

[9] D. Farley, Constructions of Evc and Efbc for groups acting on cat(0) spaces.arXiv:0902.1355v1.

[10] F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer.Math. Soc. 6 (1993), no. 2, 249–297. MR1179537 (93h:57032)

[11] D. Juan-Pineda and I. J. Leary, On classifying spaces for the family of virtually cyclicsubgroups, in Recent developments in algebraic topology, 135–145, Contemp. Math.,407, Amer. Math. Soc., Providence, RI. MR2248975 (2007d:19001)

[12] W. Luck, Transformation groups and algebraic K-theory, Lecture Notes in Math., 1408,Springer, Berlin, 1989. MR1027600 (91g:57036)

[13] W. Luck, Survey on classifying spaces for families of subgroups, in Infinite groups: geo-metric, combinatorial and dynamical aspects, 269–322, Progr. Math., 248, Birkhauser,Basel. MR2195456 (2006m:55036)

[14] W. Luck and H. Reich, The Baum-Connes and the Farrell-Jones conjectures in K- andL-theory, in Handbook of K-theory. Vol. 1, 2, 703–842, Springer, Berlin. MR2181833(2006k:19012)

[15] W. Luck and M. Weiermann, On the classifying space of the family of virtually cyclicsubgroups. To appear in the Proceedings in honour of Farrell and Jones in Pure andApplied Mathematic Quarterly. Preprintreihe SFB 478 — Geometrische Strukturen inder Mathematik, Heft 453, Munster, arXiv:math.AT/0702646v2.

[16] G. Mislin and A. Valette, Proper group actions and the Baum-Connes conjecture,Birkhauser, Basel, 2003. MR2027168 (2005d:19007)

[17] J. R. Munkres, Topology: a first course, Prentice Hall, Englewood Cliffs, N.J., 1975.MR0464128 (57 #4063)

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[19] T. tom Dieck, Transformation groups, de Gruyter, Berlin, 1987. MR0889050 (89c:57048)


214 Wolfgang Luck

Received February 3, 2009; accepted March 16, 2009

Wolfgang LuckWestfalische Wilhelms-Universitat Munster, Mathematisches InstitutEinsteinstr. 62, D-48149 Munster, GermanyE-mail: [email protected]

URL: http://www.math.uni-muenster.de/u/lueck/




C∗-algebras over topological spaces:

the bootstrap class

Ralf Meyer and Ryszard Nest



Abstract. We carefully define and study C∗-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. Weexplain the triangulated category structure on the bivariant Kasparov theory over a topo-logical space and study the analogue of the bootstrap class for C∗-algebras over a finitetopological space.

1. Introduction

If X is a locally compact Hausdorff space, then there are various equivalentcharacterizations of what it means for X to act on a C∗-algebra A. Themost common definition uses an essential ∗-homomorphism from C0(X) to thecenter of the multiplier algebra of A. An action of this kind is equivalent to acontinuous map from the primitive ideal space Prim(A) of A to X . This makessense in general: A C∗-algebra over a topological space X , which may be non-Hausdorff, is a pair (A,ψ), where A is a C∗-algebra and ψ : Prim(A)→ X is acontinuous map. One of the purposes of this article is to discuss this definitionand relate it to other notions due to Eberhard Kirchberg and Alexander Bonkat[10, 4].

An analogue of Kasparov theory for C∗-algebras over locally compact Haus-dorff spaces was defined already by Gennadi Kasparov in [9]. He used it in hisproof of the Novikov conjecture for subgroups of Lie groups. Kasparov’s defi-nition was extended by Eberhard Kirchberg to the non-Hausdorff case in [10],in order to generalize classification results for simple, purely infinite, nuclearC∗-algebras to the non-simple case. In his thesis [4], Alexander Bonkat studies

Ralf Meyer was supported by the German Research Foundation (Deutsche Forschungs-

gemeinschaft (DFG)) through the Institutional Strategy of the University of Gottingen.

216 Ralf Meyer and Ryszard Nest

an even more general theory and extends the basic results of Kasparov theoryto this setting.

This article is part of an ongoing project to compute the Kasparov groupsKK∗(X ;A,B) for a topological space X and C∗-algebras A and B overX . Theaim is a Universal Coefficient Theorem in this context that is useful for theclassification programme. At the moment, we can achieve this goal for somefinite topological spaces (see [16]), but the general situation, even in the finitecase, is unclear.

Here we describe an analogue of the bootstrap class for C∗-algebras overa topological space. Although we also propose a definition for infinite spacesin §4.4, most of our results are limited to finite spaces.

Our first task is to clarify the definition of C∗-algebras over X ; this is themain point of Section 2. Our definition is quite natural, but more restric-tive than the definitions in [10, 4]. The approach there is to use the mapO(X) → O(PrimA) induced by ψ : Prim(A) → X , where O(X) denotes thecomplete lattice of open subsets of X . If X is a sober space—this is a verymild assumption that is also made under a different name in [10, 4]—then wecan recover it from the lattice O(X), and a continuous map Prim(A) → X isequivalent to a map O(X)→ O(PrimA) that commutes with arbitrary unionsand finite intersections.

The definition of the Kasparov groups KK∗(X ;A,B) still makes sense forany map O(X)→ O(PrimA) (in the category of sets), that is, even the restric-tions imposed in [10, 4] can be removed. But such a map O(X)→ O(PrimA)corresponds to a continuous map Prim(A)→ Y for another, more complicatedspace Y that contains X as a subspace. Hence the definitions in [10, 4] are,in fact, not more general. But they complicate computations because the dis-continuities add further input data which must be taken into account even forexamples where they vanish because the action is continuous.

Since the relevant point-set topology is widely unknown among operator al-gebraists, we also recall some basic notions such as sober spaces and Alexandrovspaces. The latter are highly non-Hausdorff spaces—-Alexandrov T1-spaces arenecessarily discrete—which are essentially the same as preordered sets. Anyfinite topological space is an Alexandrov space, and their basic properties arecrucial for this article. To get acquainted with the setup, we simplify thedescription of C∗-algebras over Alexandrov spaces and discuss some small ex-amples. These rather elementary considerations appeared previously in thetheory of locales.

In Section 3, we briefly recall the definition and the basic properties ofbivariant Kasparov theory for C∗-algebras over a topological space. We omitmost proofs because they are similar to the familiar arguments for ordinaryKasparov theory and because the technical details are already dealt with in [4].We emphasize the triangulated category structure on the Kasparov categoryover X because it plays an important role in connection with the bootstrapclass.


C∗-algebras over topological spaces: the bootstrap class 217

In Section 4, we define the bootstrap class over a topological space X . If Xis finite, we give criteria for a C∗-algebra over X to belong to the bootstrapclass. These depend heavily on the relation between Alexandrov spaces andpreordered sets and therefore do not extend directly to infinite spaces.

We define the X-equivariant bootstrap class B(X) as the localizing subcat-egory of the Kasparov category of C∗-algebras over X that is generated bythe basic objects (C, x) for x ∈ X , where we identify x ∈ X with the corre-sponding constant map Prim(C) → X . Notice that this is exactly the list ofall C∗-algebras over X with underlying C∗-algebra C.

We show that a nuclear C∗-algebra (A,ψ) over X is in the X-equivariantbootstrap class if and only if its “fibers” A(x) belong to the usual bootstrapclass for all x ∈ X . These fibers are certain subquotients ofA; if ψ : Prim(A)→X is a homeomorphism, then they are exactly the simple subquotients of theC∗-algebra A.

The bootstrap class we define is the class of objects where we expect aUniversal Coefficient Theorem to hold. If A and B belong to the bootstrapclass, then an element of KK∗(X ;A,B) is invertible if and only if it is fiberwiseinvertible on K-theory, that is, the induced maps K∗

(A(x)

)→ K∗

(B(x)

)are

invertible for all x ∈ X . This follows easily from our definition of the bootstrapclass. The proof of our criterion for a C∗-algebra over X to belong to the boot-strap class already provides a spectral sequence that computes KK∗(X ;A,B)in terms of nonequivariant Kasparov groups. Unfortunately, this spectral se-quence is not useful for classification purposes because it rarely degenerates toan exact sequence.

We call a C∗-algebra over X tight if the map Prim(A) → X is a home-omorphism. This implies that its fibers are simple. We show in Section 5that any separable nuclear C∗-algebra over X is KK(X)-equivalent to a tight,separable, nuclear, purely infinite, stable C∗-algebra over X . The main issueis tightness. By Kirchberg’s classification result, this model is unique up toX-equivariant ∗-isomorphism. In this sense, tight, separable, nuclear, purelyinfinite, stable C∗-algebras over X are classified up to isomorphism by the iso-morphism classes of objects in a certain triangulated category: the subcategoryof nuclear C∗-algebras over X in the Kasparov category. The difficulty is toreplace this complete “invariant” by a more tractable one that classifies objectsof the—-possibly smaller—bootstrap category B(X) by K-theoretic data.

If C is a category, then we write A ∈∈ C to denote that A is an objectof C—as opposed to a morphism in C.

2. C∗-algebras over a topological space

We define the category C∗alg(X) of C∗-algebras over a topological space X .In the Hausdorff case, this amounts to the familiar category of C0(X)-C∗-algebras. For non-Hausdorff spaces, our notion is related to another one byEberhard Kirchberg. For the Universal Coefficient Theorem, we must add somecontinuity conditions to Kirchberg’s definition of C∗alg(X). We explain in §2.9



why these conditions result in essentially no loss of generality. Furthermore,we explain briefly why it is allowed to restrict to the case where the underlyingspaceX is sober, and we consider some examples, focusing on special propertiesof finite spaces and Alexandrov spaces.

2.1. The Hausdorff case. Let A be a C∗-algebra and let X be a locallycompact Hausdorff space. There are various equivalent additional structureson A that turn it into a C∗-algebra over X (see [17] for the proofs of mostof the following assertions). The most common definition is the following onefrom [9]:

Definition 2.1. A C0(X)-C∗-algebra is a C∗-algebra A together with an es-sential ∗-homomorphism ϕ from C0(X) to the center of the multiplier algebraof A. We abbreviate h · a := ϕ(h) · a for h ∈ C0(X).

A ∗-homomorphism f : A → B between two C0(X)-C∗-algebras is C0(X)-linear if f(h · a) = h · f(a) for all h ∈ C0(X), a ∈ A.

Let C∗alg(C0(X)

)be the category of C0(X)-C∗-algebras, whose morphisms

are the C0(X)-linear ∗-homomorphisms.

A map ϕ as above is equivalent to an A-linear essential ∗-homomorphism

ϕ : C0(X,A) ∼= C0(X)⊗max A→ A, f ⊗ a 7→ ϕ(f) · a,

which exists by the universal property of the maximal tensor product; thecentrality of ϕ ensures that ϕ is a ∗-homomorphism and well-defined. Con-versely, we get ϕ back from ϕ by restricting to elementary tensors; the as-sumed A-linearity of ϕ ensures that ϕ(h) · a := ϕ(h ⊗ a) is a multiplier of A.The description via ϕ has two advantages: it requires no multipliers, and theresulting class in KK0(C0(X,A), A) plays a role in connection with duality inbivariant Kasparov theory (see [8]).

Any C0(X)-C∗-algebra is isomorphic to the C∗-algebra of C0(X)-sections ofan upper semi-continuous C∗-algebra bundle overX (see [17]). Even more, thisyields an equivalence of categories between C∗alg

(C0(X)

)and the category of

upper semi-continuous C∗-algebra bundles over X .

Definition 2.2. Let Prim(A) denote the primitive ideal space of A, equippedwith the usual hull-kernel topology, also called Jacobson topology.

The Dauns-Hofmann Theorem identifies the center of the multiplier algebraof A with the C∗-algebra Cb

(Prim(A)

)of bounded continuous functions on

the primitive ideal space of A. Therefore, the map ϕ in Definition 2.1 is of theform

ψ∗ : C0(X)→ Cb(PrimA), f 7→ f ψ,for some continuous map ψ : Prim(A)→ X (see [17]). Thus ϕ and ψ are equiv-alent additional structures. We use such maps ψ to generalize Definition 2.1to the non-Hausdorff case.



2.2. The general definition. Let X be an arbitrary topological space.

Definition 2.3. A C∗-algebra over X is a pair (A,ψ) where A is a C∗-algebraand is ψ a continuous map Prim(A)→ X .

Our next task is to define morphisms between C∗-algebras A and B overthe same space X . This requires some care because the primitive ideal spaceis not functorial for arbitrary ∗-homomorphisms.

Definition 2.4. For a topological spaceX , let O(X) be the set of open subsetsof X , partially ordered by ⊆.

Definition 2.5. For a C∗-algebra A, let I(A) be the set of all closed ∗-idealsin A, partially ordered by ⊆.

The partially ordered sets (O(X),⊆) and (I(A),⊆) are complete lattices,that is, any subset in them has both an infimum

∧S and a supremum

∨S.

Namely, in O(X), the supremum is⋃S, and the infimum is the interior of

⋂S;

in I(A), the infimum and supremum are∧

I∈S

I =⋂

I∈S

I,∨

I∈S

I =∑

I∈S

I.

We always identify O(Prim(A)

)and I(A) using the isomorphism

(2.6) O(Prim(A)

) ∼= I(A), U 7→⋂

p∈Prim(A)\U

p

(see [7, §3.2]). This is a lattice isomorphism and hence preserves infima andsuprema.

Let (A,ψ) be a C∗-algebra over X . We get a map

ψ∗ : O(X)→ O(PrimA) ∼= I(A), U 7→ A(U) := p ∈ Prim(A) | ψ(p) ∈ U.We usually write A(U) ∈ I(A) for the ideal and ψ∗(U) or ψ−1(U) for thecorresponding open subset of Prim(A). If X is a locally compact Hausdorffspace, then A(U) := C0(U) · A for all U ∈ O(X).

Example 2.7. For any C∗-algebra A, the pair (A, idPrimA) is a C∗-algebra overPrim(A); the ideals A(U) for U ∈ O(PrimA) are given by (2.6). C∗-algebrasover topological spaces of this form play an important role in §5, where we callthem tight.

Lemma 2.8. The map ψ∗ is compatible with arbitrary suprema (unions) andfinite infima (intersections), so that

A

( ⋃

U∈S

U

)=∑

U∈S

A(U), A

( ⋂

U∈F

U

)=⋂

U∈F

A(U)

for any subset S ⊆ O(X) and for any finite subset F ⊆ O(X).

Proof. This is immediate from the definition.



Taking for S and F the empty set, this specializes to A(∅) = 0 andA(X) = A. Taking S = U, V with U ⊆ V , this specializes to the monotonic-ity property

U ⊆ V =⇒ A(U) ⊆ A(V );

We will implicitly use later that these properties follow from compatibility withfinite infima and suprema.

The following lemma clarifies when the map ψ∗ is compatible with infiniteinfima.

Lemma 2.9. If the map ψ : Prim(A) → X is open or if X is finite, thenthe map ψ∗ : O(X) → I(A) preserves infima—that is, it maps the interior of⋂U∈S U to the ideal

⋂U∈S A(U) for any subset S ⊆ O(X). Conversely, if ψ∗

preserves infima and X is a T1-space, that is, points in X are closed, then ψis open.

Since preservation of infinite infima is automatic for finite X , the converseassertion cannot hold for general X .

Proof. If X is finite, then any subset of O(X) is finite, and there is nothingmore to prove. Suppose that ψ is open. Let V be the interior of

⋂U∈S U . Let

W ⊆ Prim(A) be the open subset that corresponds to the ideal⋂U∈S ψ

∗(U).We must show ψ∗(V ) = W . Monotonicity yields ψ∗(V ) ⊆W . Since ψ is open,ψ(W ) is an open subset of X . By construction, ψ(W ) ⊆ U for all U ∈ Sand hence ψ(W ) ⊆ V . Thus ψ∗(V ) ⊇ ψ∗

(ψ(W )

)⊇ W ⊇ ψ∗(V ), so that

ψ∗(V ) = W .Now suppose, conversely, that ψ∗ preserves infima and that points in X

are closed. Assume that ψ is not open. Then there is an open subset W inPrim(A) for which ψ(W ) is not open inX . Let S := X\x | x ∈ X\ψ(W ) ⊆O(X); this is where we need points to be closed. We have

⋂U∈S U = ψ(W )

and⋂U∈S ψ

∗(U) = ψ−1(ψ(W )

). Since ψ(W ) is not open, the infimum V

of S in O(X) is strictly smaller than ψ(W ). Hence ψ∗(V ) cannot contain W .But W is an open subset of ψ−1

(ψ(W )

)and hence contained in the infimum

of ψ∗(S) in O(PrimA). Therefore, ψ∗ does not preserve infima, contrary toour assumption. Hence ψ must be open.

For a locally compact Hausdorff space X , the map Prim(A)→ X is open ifand only if A corresponds to a continuous C∗-algebra bundle over X (see [17,Thm. 2.3]).

Definition 2.10. Let A and B be C∗-algebras over a topological space X .A ∗-homomorphism f : A → B is X-equivariant if f

(A(U)

)⊆ B(U) for all

U ∈ O(X).

For locally compact Hausdorff spaces, this is equivalent to C0(X)-linearityby the following variant of [4, Prop. 5.4.7]:

Proposition 2.11. Let A and B be C∗-algebras over a locally compact Haus-dorff space X, and let f : A→ B be a ∗-homomorphism. The following asser-tions are equivalent:



(1) f is C0(X)-linear;(2) f is X-equivariant, that is, f

(A(U)

)⊆ B(U) for all U ∈ O(X);

(3) f descends to the fibers, that is, f(A(X \ x)

)⊆ B(X \ x) for all

x ∈ X.

To understand the last condition, recall that the fibers of the C∗-algebrabundle associated to A are Ax := A/A(X \ x). Condition (3) means that fdescends to maps fx : Ax → Bx for all x ∈ X .

Proof. It is clear that (1)=⇒(2)=⇒(3). The equivalence (3) ⇐⇒ (1) is theassertion of [4, Prop. 5.4.7]. To check that (3) implies (1), take h ∈ C0(X) anda ∈ A. We get f(h · a) = h · f(a) provided both sides have the same valuesat all x ∈ X because the map A → ∏

x∈X Ax is injective. Now (3) implies

f(h · a)x = h(x) · f(a)x because(h− h(x)

)· a ∈ A(X \ x).

Definition 2.12. Let C∗alg(X) be the category whose objects are C∗-algebrasover X and whose morphisms are X-equivariant ∗-homomorphisms. We writeHomX(A,B) for this set of morphisms.

Proposition 2.11 yields an isomorphism of categories

C∗alg(C0(X)

) ∼= C∗alg(X).

In this sense, our theory for general spaces extends the more familiar theoryof C0(X)-C∗-algebras.

2.3. Locally closed subsets and subquotients.

Definition 2.13. A subset C of a topological space X is called locally closedif it is the intersection of an open and a closed subset or, equivalently, of theform C = U \ V with U, V ∈ O(X); we can also assume V ⊆ U here. We letLC(X) be the set of locally closed subsets of X .

A subset is locally closed if and only if it is relatively open in its closure.Being locally closed is inherited by finite intersections, but not by unions orcomplements.

Definition 2.14. Let X be a topological space and let (A,ψ) be a C∗-algebraover X . Write C ∈ LC(X) as C = U \ V for open subsets U, V ⊆ X withV ⊆ U . We define

A(C) := A(U) / A(V ).

Lemma 2.15. The subquotient A(C) does not depend on U and V above.

Proof. Let U1, V1, U2, V2 ∈ O(X) satisfy V1 ⊆ U1, V2 ⊆ U2, and U1 \ V1 =U2 \ V2. Then V1 ∪ U2 = U1 ∪ U2 = U1 ∪ V2 and V1 ∩ U2 = V1 ∩ V2 = U1 ∩ V2.Since U 7→ A(U) preserves unions, this implies

A(U2) +A(V1) = A(U1) +A(V2).



We divide this equation by A(V1 ∪ V2) = A(V1) +A(V2). This yields

A(U2) +A(V1)

A(V1 ∪ V2)∼= A(U2)

A(U2) ∩A(V1 ∪ V2)=

A(U2)

A(U2 ∩ (V1 ∪ V2)

) =A(U2)

A(V2)

on the left hand side and, similarly, A(U1) / A(V1) on the right hand side.Hence A(U1) / A(V1) ∼= A(U2) / A(V2) as desired.

Now assume that X = Prim(A) and ψ = idPrim(A). Lemma 2.15 associates asubquotient A(C) of A to each locally closed subset of Prim(A). Equation (2.6)shows that any subquotient of A arises in this fashion; here subquotient means:a quotient of one ideal in A by another ideal in A. Open subsets of X corre-spond to ideals, closed subsets to quotients of A. For any C ∈ LC(PrimA),there is a canonical homeomorphism Prim

(A(C)

) ∼= C. This is well-knownif C is open or closed, and the general case reduces to these special cases.

Example 2.16. If Prim(A) is a finite topological T0-space, then any single-ton p in Prim(A) is locally closed (this holds more generally for the Alexan-drov T0-spaces introduced in §2.7 and follows from the description of closedsubsets in terms of the specialization preorder).

Since PrimA(C) ∼= C, the subquotients Ap := A(p) for p ∈ Prim(A) areprecisely the simple subquotients of A.

Example 2.17. Consider the interval [0, 1] with the topology where the non-empty closed subsets are the closed intervals [a, 1] for all a ∈ [0, 1]. A nonemptysubset is locally closed if and only if it is either of the form [a, 1] or [a, b) fora, b ∈ [0, 1] with a < b. In this space, singletons are not locally closed. Hencea C∗-algebra with this primitive ideal space has no simple subquotients.

2.4. Functoriality and tensor products.

Definition 2.18. Let X and Y be topological spaces. A continuous mapf : X → Y induces a functor

f∗ : C∗alg(X)→ C∗alg(Y ), (A,ψ) 7→ (A, f ψ).

Thus X 7→ C∗alg(X) is a functor from the category of topological spaces tothe category of categories (up to the usual issues with sets and classes).

Since (f ψ)−1 = ψ−1 f−1, we have

(f∗A)(C) = A(f−1(C)

)for all C ∈ LC(Y ).

If f : X → Y is the embedding of a subset with the subspace topology, wealso write

iYX := f∗ : C∗alg(X)→ C∗alg(Y )

and call this the extension functor fromX to Y . We have (iYXA)(C) = A(C∩X)for all C ∈ LC(Y ).

Definition 2.19. Let X be a topological space and let Y be a locally closedsubset of X , equipped with the subspace topology. Let (A,ψ) be a C∗-algebraover X . Its restriction to Y is a C∗-algebra A|Y over Y , consisting of the



C∗-algebra A(Y ) defined as in Definition 2.14, equipped with the canonicalmap

PrimA(Y )∼=−→ ψ−1(Y )

ψ−→ Y.

Thus A|Y (C) = A(C) for C ∈ LC(Y ) ⊆ LC(X).

It is clear that the restriction to Y provides a functor

rYX : C∗alg(X)→ C∗alg(Y )

that satisfies rZY rYX = rZX if Z ⊆ Y ⊆ X and rXX = id.If Y and X are Hausdorff and locally compact, then a continuous map

f : Y → X also induces a pull-back functor

f∗ : C∗alg(X) ∼= C∗alg(C0(X)

)→ C∗alg

(C0(Y )

) ∼= C∗alg(Y ),

A 7→ C0(Y )⊗C0(X) A.

If f is the constant map Y → ⋆, the functor f∗ maps a C∗-algebra A tof∗(A) := C0(Y,A) with the obvious C0(Y )-C∗-algebra structure. This func-tor has no analogue for a non-Hausdorff space Y . Therefore, a continuousmap f : Y → X need not induce a functor f∗ : C∗alg(X) → C∗alg(Y ). Forembeddings of locally closed subsets, the functor rYX plays the role of f∗.

Lemma 2.20. Let X be a topological space and let Y ⊆ X.

(a) If Y is open, then there are natural isomorphisms

HomX(iXY (A), B) ∼= HomY

(A, rYX(B)

)

if A and B are C∗-algebras over Y and X, respectively.In other words, iXY is left adjoint to rYX .

(b) If Y is closed, then there are natural isomorphisms

HomY (rYX(A), B) ∼= HomX

(A, iXY (B)

)

if A and B are C∗-algebras over X and Y , respectively.In other words, iXY is right adjoint to rYX .

(c) For any locally closed subset Y ⊆ X, we have rYX iXY (A) = A for allC∗-algebras A over Y .

Proof. We first prove (a). We have iXY (A)(U) = A(U ∩ Y ) for all U ∈ O(X),and this is an ideal in A(U). A morphism ϕ : iXY (A) → B is equivalent toa ∗-homomorphism ϕ : A(Y ) → B(X) that maps A(U ∩ Y ) → B(U) for allU ∈ O(X). This holds for all U ∈ O(X) once it holds for U ∈ O(Y ) ⊆ O(X).Hence ϕ is equivalent to a ∗-homomorphism ϕ′ : A(Y ) → B(Y ) that mapsA(U) → B(U) for all U ∈ O(Y ). The latter is nothing but a morphismA→ rYX(B). This proves (a).

Now we turn to (b). Again, we have iXY (B)(U) = B(U∩Y ) for all U ∈ O(X),but now this is a quotient of B(U). A morphism ϕ : A→ iXY (B) is equivalentto a ∗-homomorphism ϕ : A(X)→ B(Y ) that maps A(U)→ B(U ∩ Y ) for allU ∈ O(X). Hence A(X \ Y ) is mapped to B(∅) = 0, so that ϕ descends to amap ϕ′ from A /A(X \ Y ) ∼= A(Y ) to B(Y ) that maps A(U ∩ Y ) to B(U) for



all U ∈ O(X). The latter is equivalent to a morphism rYX(A)→ B as desired.This finishes the proof of (b).

Assertion (c) is trivial.

Example 2.21. For each x ∈ X , we get a map ix = iXx : ⋆ ∼= x ⊆ X fromthe one-point space to X . The resulting functor C∗alg → C∗alg(X) maps aC∗-algebra A to the C∗-algebra ix(A) = (A, x) over X , where x also denotesthe constant map

x : Prim(A)→ X, p 7→ x for all p ∈ Prim(A).

If C ∈ LC(X), then

ix(A)(C) =

A if x ∈ C;

0 otherwise.

The functor ix plays an important role if X is finite. The generators of thebootstrap class are of the form ix(C). Each C∗-algebra over X carries a canon-ical filtration whose subquotients are of the form ix(A).

Lemma 2.22. Let X be a topological space and let x ∈ X. Then

HomX

(A, iXx (B)

) ∼= Hom(A(x), B)

for all A ∈∈ C∗alg(X), B ∈∈ C∗alg, and

HomX(iXx (A), B) ∼= Hom(A,

⋂

U∈Ux

B(U)).

for all A ∈∈ C∗alg, B ∈∈ C∗alg(X), where Ux denotes the open neighborhoodfilter of x in X. If x has a minimal open neighborhood Ux, then this becomes

HomX(iXx (A), B) ∼= Hom(A,B(Ux)

).

Recall that A ∈∈ C means that A is an object of C.

Proof. Let C := x. Then any nonempty open subset V ⊆ C contains x, sothat iCx (B)(V ) = B. This implies HomC(A, iCx (B)) ∼= Hom

(A(C), B

). Com-

bining this with iXx = iXC iCx and the adjointness relation in Lemma 2.20.(b)yields

HomX

(A, iXx (B)

) ∼= HomC

(rCX(A), iCx (B)

) ∼= Hom(A(C), B).

An X-equivariant ∗-homomorphism iXx A→ B restricts to a family of com-patible maps A = (iXx A)(U) → B(U) for all U ∈ Ux, so that we get a∗-homomorphism from A to

⋂U∈Ux

B(U). Conversely, such a ∗-homomorphism

A → ⋂U∈Ux

B(U) provides an X-equivariant ∗-homomorphism iXx A → B.This yields the second assertion.

Let A and B be C∗-algebras and let A ⊗ B be their minimal (or spatial)C∗-tensor product. Then there is a canonical continuous map

Prim(A)× Prim(B)→ Prim(A⊗B).



Therefore, if A and B are C∗-algebras over X and Y , respectively, then A⊗Bis a C∗-algebra over X × Y . This defines a bifunctor

⊗ : C∗alg(X)× C∗alg(Y )→ C∗alg(X × Y ).

In particular, if Y = ⋆ is the one-point space, then we get endofunctors⊗B on C∗alg(X) for B ∈∈ C∗alg because X × ⋆ ∼= X .If X is a Hausdorff space, then the diagonal in X ×X is closed and we get

an internal tensor product functor ⊗X in C∗alg(X) by restricting the externaltensor product in C∗alg(X×X) to the diagonal. This operation has no analoguefor general X .

2.5. Restriction to sober spaces. A space is sober if and only if it canbe recovered from its lattice of open subsets. Any topological space can becompleted to a sober space with the same lattice of open subsets. Therefore,it usually suffices to study C∗-algebras over sober topological spaces.

Definition 2.23. A topological space is sober if each irreducible closed subsetof X is the closure x of exactly one singleton of X . Here an irreducible closedsubset of X is a nonempty closed subset of X which is not the union of twoproper closed subsets of itself.

If X is not sober, let X be the set of all irreducible closed subsets of X .There is a canonical map ι : X → X which sends a point x ∈ X to its closure.If S ⊆ X is closed, let S ⊆ X be the set of all A ∈ X with A ⊆ S. The mapS 7→ S commutes with finite unions and arbitrary intersections; in particular,it maps X itself to all of X and ∅ to ∅ = ∅. Hence the subsets of X of theform S for closed subsets S ⊆ X form the closed subsets of a topology on X .

The map ι induces a bijection between the families of closed subsets of Xand X. Hence ι is continuous, and it induces a bijection ι∗ : O(X)→ O(X). It

also follows that X is a sober space because X and X have the same irreducibleclosed subsets.

Since the morphisms in C∗alg(X) only use O(X), the functor

ι∗ : C∗alg(X)→ C∗alg(X)

is fully faithful. Therefore, we do not lose much if we assume our topologicalspaces to be sober.

The following example shows a pathology that can occur if the separationaxiom T0 fails:

Example 2.24. Let X carry the chaotic topology O(X) = ∅, X. Then

X = ⋆ is the space with one point. By definition, an action of X on aC∗-algebra A is a map Prim(A) → X . But for a ∗-homomorphism A → Bbetween two C∗-algebras over X , the X-equivariance condition imposes no re-striction. Hence all maps Prim(A)→ X yield isomorphic objects of C∗alg(X).

Lemma 2.25. If X is a sober topological space, then there is a bijective cor-respondence between continuous maps Prim(A)→ X and maps O(X)→ I(A)



that commute with arbitrary suprema and finite infima; it sends a continuousmap ψ : Prim(A)→ X to the map

ψ∗ : O(X)→ O(Prim(A)

)= I(A).

Proof. We have already seen that a continuous map ψ : Prim(A) → X gener-ates a map ψ∗ with the required properties for any space X .

Conversely, let ψ∗ : O(X)→ I(A) be a map that preserves arbitrary unionsand finite intersections. Given p ∈ Prim(A), let Up be the union of all U ∈O(X) with p /∈ ψ∗(U). Then p /∈ ψ∗(Up) because ψ∗ preserves unions, and Up isthe maximal open subset with this property. Thus Ap := X \Up is the minimalclosed subset with p /∈ ψ∗(X \Ap). This subset is nonempty because ψ∗(X) =Prim(A) contains p, and irreducible because ψ∗ preserves finite intersections.

Since X is sober, there is a unique ψ(p) ∈ X with Ap = ψ(p). Thisdefines a map ψ : Prim(A)→ X . If U ⊆ X is open, then ψ(p) /∈ U if and onlyif Ap ∩ U = ∅, if and only if p /∈ ψ∗(U). Hence ψ∗(U) = ψ−1(U). This showsthat ψ is continuous and generates ψ∗. Thus the map ψ → ψ∗ is surjective.

Since sober spaces are T0, two different continuous maps ψ1, ψ2 : Prim(A)→X generate different maps ψ∗1 , ψ

∗2 : O(X) → I(A). Hence the map ψ → ψ∗ is

also injective.

2.6. Some very easy examples. Here we describe C∗-algebras over the threesober topological spaces with at most two points.

Example 2.26. If X is a single point, then C∗alg(X) is isomorphic to the cat-egory of C∗-algebras (without any extra structure).

Up to homeomorphism, there are two sober topological spaces with twopoints. The first one is the discrete space.

Example 2.27. The category of C∗-algebras over the discrete two-point spaceis equivalent to the product category C∗alg× C∗alg of pairs of C∗-algebras.

More generally, if X = X1 ⊔X2 is a disjoint union of two subspaces, then

(2.28) C∗alg(X1 ⊔X2) ≃ C∗alg(X1)× C∗alg(X2).

Thus it usually suffices to study connected spaces.

Example 2.29. Another sober topological space with two points is X = 1, 2with

O(X) =∅, 1, 1, 2

.

A C∗-algebra over this space comes with a single distinguished ideal A(1) ⊳ A,which is arbitrary. Thus we get the category of pairs (I, A) where I is an idealin A. We may associate to this data the C∗-algebra extension I A ։ A/I.In fact, the morphisms in HomX(A,B) are the morphisms of extensions

A(1)

// // A

// // A / A(1)

B(1) // // B // // B / B(1).



Thus C∗alg(X) is equivalent to the category of C∗-algebra extensions. Thisexample is also studied in [4].

2.7. Topologies and partial orders. Certain non-Hausdorff spaces are veryclosely related to partially ordered sets. In particular, there is a bijectionbetween sober topologies and partial orders on a finite set. Here we recall therelevant constructions.

Definition 2.30. Let X be a topological space. The specialization preorder on X is defined by x y if the closure of x is contained in the closure of yor, equivalently, if y is contained in all open subsets of X that contain x. Twopoints x and y are called topologically indistinguishable if x y and y x,that is, the closures of x and y are equal.

The separation axiom T0 means that topologically indistinguishable pointsare equal. Since this is automatic for sober spaces, is a partial order on Xin all cases we need. As usual, we write x ≺ y if x y and x 6= y, and x yand x ≻ y are equivalent to y x and y ≺ x, respectively.

The separation axiom T1 requires points to be closed. This is equivalent tothe partial order being trivial, that is, x y if and only if x = y. Thus ourpartial order is only meaningful for highly nonseparated spaces.

The following notion goes back to an article by Paul Alexandrov from 1937([1]); see also [2] for a more recent reference, or the English Wikipedia entryon the Alexandrov topology.

Definition 2.31. Let (X,≤) be a preordered set. A subset S ⊆ X is calledAlexandrov-open if S ∋ x ≤ y implies y ∈ S. The Alexandrov-open subsetsform a topology on X called the Alexandrov topology.

A subset of X is closed in the Alexandrov topology if and only if S ∋ xand x ≥ y imply S ∋ y. It is locally closed if and only if it is convex, thatis, x ≤ y ≤ z and x, z ∈ S imply y ∈ S. In particular, singletons are locallyclosed (compare Example 2.16).

The specialization preorder for the Alexandrov topology is the given pre-order. Moreover, a map (X,≤) → (Y,≤) is continuous for the Alexandrovtopology if and only if it is monotone. Thus we have identified the category ofpreordered sets with monotone maps with a full subcategory of the categoryof topological spaces.

It also follows that if a topological space carries an Alexandrov topology forsome preorder, then this preorder must be the specialization preorder. In thiscase, we call the space an Alexandrov space or a finitely generated space. Thefollowing lemma provides some equivalent descriptions of Alexandrov spaces;the last two explain in what sense these spaces are finitely generated.

Lemma 2.32. Let X be a topological space. The following are equivalent:

• X is an Alexandrov space;• an arbitrary intersection of open subsets of X is open;• an arbitrary union of closed subsets of X is closed;



• every point of X has a smallest neighborhood;• a point x lies in the closure of a subset S of X if and only if x ∈ y

for some y ∈ S;• X is the inductive limit of the inductive system of its finite subspaces.

Corollary 2.33. Any finite topological space is an Alexandrov space. Thusthe construction of Alexandrov topologies and specialization preorders providesa bijection between preorders and topologies on a finite set.

Definition 2.34. Let X be an Alexandrov space. We denote the minimalopen neighborhood of x ∈ X by Ux ∈ O(X).

We have

Ux ⊆ Uy ⇐⇒ x ∈ Uy ⇐⇒ y ∈ x ⇐⇒ y ⊆ x ⇐⇒ y x.If X is a sober Alexandrov space, then we can simplify the data for a

C∗-algebra over X as follows:

Lemma 2.35. A C∗-algebra over a sober Alexandrov space X is determineduniquely by a C∗-algebra A together with ideals A(Ux)⊳A for all x ∈ X, subject

to the two conditions∑x∈X A(Ux) = A and

(2.36) A(Ux) ∩A(Uy) =∑

z∈Ux∩Uy

A(Uz) for all x, y ∈ X.

Proof. A map O(X) → I(A) that preserves suprema and maps Ux to A(Ux)

for all x ∈ X must map U =∨x∈U Ux to

∨x∈U A(Ux) =

∑x∈U A(Ux). The

map so defined preserves suprema by construction. The two hypotheses of thelemma ensure A(X) = A and A(Ux ∩ Uy) = A(Ux) ∩ A(Uy) for all x, y ∈ X .Hence they are necessary for preservation of finite infima.

Since the lattice I(A) ∼= O(PrimA) is distributive, (2.36) implies

A(U) ∧A(V ) =∨

x∈U

A(Ux) ∧∨

y∈V

A(Vy) =∨

(x,y)∈U×V

A(Ux) ∧A(Vy)

=∨

(x,y)∈U×V

A(Ux ∩ Vy) = A(U ∩ V );

the last step uses that U 7→ A(U) commutes with suprema. We clearly haveA(∅) = 0 as well, so that U 7→ A(U) preserves arbitrary finite intersections.Therefore, our map O(X)→ I(A) satisfies the conditions in Lemma 2.25 andhence comes from a continuous map PrimA→ X .

Of course, a ∗-homomorphism A → B between two C∗-algebras over X isX-equivariant if and only if it maps A(Ux)→ B(Ux) for all x ∈ X .

Equation (2.36) implies A(Ux) ⊆ A(Uy) if Ux ⊆ Uy, that is, if x y. Thusthe map x 7→ A(Ux) is order-reversing. It sometimes happens that Ux∩Uy = Uzfor some x, y, z ∈ X . In this case, we may drop the ideal A(Uz) from thedescription of a C∗-algebra over X and replace the condition (2.36) for x, y byA(Uw) ⊆ A(Ux) ∩A(Uy) for all w ∈ Ux ∩ Uy.



2.8. Some more examples. A useful way to represent finite partially orderedsets and hence finite sober topological spaces is via finite directed acyclic graphs.

To a partial order on X , we associate the finite directed acyclic graphwith vertex set X and with an arrow x ← y if and only if x ≺ y and thereis no z ∈ X with x ≺ z ≺ y. We can recover the partial order from thisgraph by letting x y if and only if the graph contains a directed pathx← x1 ← · · · ← xn ← y.

We have reversed arrows here because an arrow x→ y means that A(Ux) ⊆A(Uy). Furthermore, x ∈ Uy if and only if there is a directed path from x to y.Thus we can read the meaning of the relations (2.36) from the graph.

Example 2.37. Let (X,≥) be a set with a total order, such as 1, . . . , n withthe order ≥. The corresponding graph is

1 // 2 // 3 // · · · // n.

For totally ordered X , (2.36) is equivalent to monotonicity of the map x 7→A(Ux) with respect to the opposite order ≤ on X . As a consequence, aC∗-algebra over X is nothing but a C∗-algebra A together with a mono-tone map (X,≤) → I(A), x 7→ A(Ux), such that

∨x∈X A(Ux) = A. For

X =(1, . . . , n,≥

), the latter condition just means A(Un) = A, so that we

can drop this ideal. Thus we get C∗-algebras with an increasing chain of n− 1ideals I1 ⊳ I2 ⊳ · · · ⊳ In−1 ⊳ A. This situation is studied in detail in [16].

Using that any finite topological space is an Alexandrov space, we can easilylist all homeomorphism classes of finite topological spaces with, say, three orfour elements. We only consider sober spaces here, and we assume connected-ness to further reduce the number of cases. Under these assumptions, Figure 1contains a complete list. The first and fourth case are already contained inExample 2.37. Lemma 2.35 describes C∗-algebras over the spaces in Figure 1as C∗-algebras equipped with three or four ideals A(Ux) for x ∈ X , subject tosome conditions, which often make some of the ideals redundant.

Example 2.38. The second graph in Figure 1 describes C∗-algebras with threeideals A(Uj), j = 1, 2, 3, subject to the conditions A(U2)∩A(U3) = A(U1) andA(U2) + A(U3) = A. This is equivalent to prescribing only two ideals A(U2)and A(U3) subject to the single condition A(U2) +A(U3) = A.

Example 2.39. Similarly, the third graph in Figure 1 describes C∗-algebras withtwo distinguished ideals A(U1) and A(U2) subject to the condition A(U1) ∩A(U2) = 0; here U3 = X implies A(U3) = A.

Example 2.40. The ninth graph in Figure 1 is more complicated. We label ourpoints by 1, 2, 3, 4 such that 1 → 3 ← 2 → 4. Here we have a C∗-algebra Awith four ideals Ij := A(Uj) for j = 1, 2, 3, 4, subject to the conditions

I1 ⊆ I3, I1 ∩ I4 = 0, I2 = I3 ∩ I4, I3 + I4 = A.

Thus the ideal I2 is redundant, and we are left with three ideals I1, I3, I4subject to the conditions I1 ⊆ I3, I1 ∩ I4 = 0, and I3 + I4 = A.



• // • // • • //

@@@

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•

•

• // •

•

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• // • // • // • • // • //

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•

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Figure 1. Connected directed acyclic graphs with three orfour vertices

2.9. How to treat discontinuous bundles. The X-equivariant Kasparovtheory constructed in [10, 4] works for any map ψ∗ : O(X) → I(A), we donot need the conditions in Lemma 2.25. Here we show how to reduce thismore general situation to the case considered above: discontinuous actions ofO(X) as in [10, 4] are equivalent to continuous actions of another space Ythat contains X as a subspace. The category C∗alg(Y ) contains C∗alg(X) asa full subcategory, and a similar statement holds for the associated Kasparovcategories. As a result, allowing general maps ψ∗ merely amounts to replacingthe space X by the larger space Y . For C∗-algebras that really live overthe subspace X , the extension to Y significantly complicates the computationof the Kasparov groups. This is why we always require ψ∗ to satisfy theconditions in Lemma 2.25, which ensure that it comes from a continuous mapPrim(A)→ X .

Example 2.41. Let X = 1, 2 with the discrete topology. A monotone mapψ∗ : O(X) → A with ψ∗(∅) = 0 and ψ∗(X) = A as considered in [4, 10] isequivalent to specifying two arbitrary ideals A(1) and A(2). This automaticallygenerates the ideals A(1) ∩ A(2) and A(1) ∪ A(2). We can encode these fourideals in an action of a topological space Y with four points 1∩ 2, 1, 2, 3 andopen subsets

∅, 1 ∩ 2, 1 ∩ 2, 1, 1 ∩ 2, 2, 1 ∩ 2, 1, 2, 1 ∩ 2, 1, 2, 3.



The corresponding graph is the seventh one in Figure 1. The map ψ∗ mapsthese open subsets to the ideals

0, A(1) ∩A(2), A(1), A(2), A(1) ∪A(2), A,

respectively. This defines a complete lattice morphism O(Y )→ I(A), and anycomplete lattice morphism is of this form for two ideals A(1) and A(2). Thusan action of O(1, 2) in the generalized sense considered in [4, 10] is equivalentto an action of Y in our sense.

Any X-equivariant ∗-homomorphism A→ B between two such discontinu-ous C∗-algebras overX will also preserve the ideals A(1)∩A(2) and A(1)∪A(2).Hence it is Y -equivariant as well. Therefore, the above construction providesan equivalence of categories between C∗alg(Y ) and the category of C∗-algebraswith an action of O(X) in the sense of [4, 10].

Whereas the computation of

KK∗(X ;A,B) ∼= KK∗(A(1), B(1)

)×KK∗

(A(2), B(2)

)

for two C∗-algebras A and B over X is trivial, the corresponding problem forC∗-algebras over Y is an interesting problem: this is one of the small exampleswhere filtrated K-theory does not yet suffice for classification.

This simple example generalizes as follows. Let

f : O(X)→ I(A) ∼= O(PrimA)

be an arbitrary map. Let Y := 2O(X) be the power set of O(X), partiallyordered by inclusion. We describe the topology on Y below. We embed theoriginal space X into Y by mapping x ∈ X to its open neighborhood filter:

U : X → Y, x 7→ U ∈ O(X) | x ∈ U.We define a map

ψ : Prim(A)→ Y, p 7→ U ∈ O(X) | p ∈ f(U).For y ∈ Y , let Y⊇y := x ∈ Y | x ⊇ y. For a singleton U with U ∈ O(X),we easily compute

ψ−1(Y⊇U) = f(U) ∈ I(A) ∼= O(PrimA).

Moreover, Y⊇y∪z = Y⊇y ∩ Y⊇z, so that we get

ψ−1(Y⊇U1,...,Un) = f(U1) ∩ · · · ∩ f(Un).

A similar argument shows that

U−1(Y⊇U1,...,Un) = U−1(Y⊇U1) ∩ · · · ∩ U−1(Y⊇Un) = U1 ∩ · · · ∩ Un.We equip Y with the topology that has the sets Y⊇F for finite subsets F

of O(X) as a basis. It is clear from the above computations that this makesthe maps ψ and U continuous; even more, the subspace topology on the rangeof U is the given topology on X .

As a consequence, any map f : O(X) → I(A) turns A into a C∗-algebraover the space Y ⊇ X . Conversely, given a C∗-algebra over Y , we define



f : O(X) → I(A) by f(U) := ψ−1(Y⊇U). This construction is inverse tothe one above. Furthermore, a ∗-homomorphism A→ B that maps fA(U) tofB(U) for all U ∈ O(X) also maps ψ∗A(U) to ψ∗B(U) for all U ∈ O(Y ). We cansum this up as follows:

Theorem 2.42. The category of C∗-algebras equipped with a map f : O(X)→I(A) is isomorphic to the category of C∗-algebras over Y .

If f has some additional properties like monotonicity, or is a lattice mor-phism, then this limits the range of the map ψ above and thus allows us toreplace Y by a smaller subset. In [10, Def. 1.3] and [4, Def. 5.6.2], an actionof a space X on a C∗-algebra is defined to be a map f : O(X) → I(A) thatis monotone and satisfies f(∅) = 0 and f(X) = A. These assumptions areequivalent to

U ∈ ψ(p), U ⊆ V =⇒ V ∈ ψ(p)

and ∅ /∈ ψ(p) and X ∈ ψ(p) for all p ∈ Prim(A). Hence the category ofC∗-algebras with an action of X in the sense of [4, 10] is equivalent to thecategory of C∗-algebras over the space

Y ′ := y ⊆ O(X) | y ∋ U ⊆ V =⇒ V ∈ y, ∅ /∈ y, X ∈ y,equipped with the subspace topology from Y .

3. Bivariant K-theory for C∗-algebras over topological spaces

Let X be a topological space. Eberhard Kirchberg [10] and AlexanderBonkat [4] define Kasparov groups KK∗(X ;A,B) for separable C∗-algebrasA and B over X . More precisely, instead of a continuous map Prim(A) → Xthey use a separable C∗-algebra A with a monotone map ψ∗ : O(X) → I(A)with A(∅) = 0 and A(X) = A. This is more general because any continuousmap Prim(A)→ X generates such a map ψ∗ : O(X)→ I(A). Hence their def-initions apply to C∗-algebras over X in our sense. We have explained in §2.9why the setting in [10, 4] is, despite appearances, not more general than oursetting.

If X is Hausdorff and locally compact, KK∗(X ;A,B) agrees with GennadiKasparov’s theory RKK∗(X ;A,B) defined in [9]. In this section, we recallthe definition and some basic properties of the functor KK∗(X ;A,B) and theresulting category KK(X), and we equip the latter with a triangulated categorystructure.

3.1. The definition. We assume from now on that the topology on X has acountable basis, and we restrict attention to separable C∗-algebras.

Definition 3.1. A C∗-algebra (A,ψ) overX is called separable if A is a separa-ble C∗-algebra. Let C∗sep(X) ⊆ C∗alg(X) be the full subcategory of separableC∗-algebras over X .

To describe the cycles for KK∗(X ;A,B) recall that the usual Kasparovcycles for KK∗(A,B) are of the form (ϕ,HB, F, γ) in the even case (for KK0)and (ϕ,HB , F ) in the odd case (for KK1), where



• HB is a right Hilbert B-module;• ϕ : A→ B(HB) is a ∗-representation;• F ∈ B(HB);• ϕ(a)(F 2 − 1), ϕ(a)(F − F ∗), and [ϕ(a), F ] are compact for all a ∈ A;• in the even case, γ is a Z/2-grading on HB—that is, γ2 = 1 andγ = γ∗—that commutes with ϕ(A) and anti-commutes with F .

The following definition of X-equivariant bivariant K-theory is equivalentto the ones in [10, 4], see [10, Def. 4.1], and [4, Def. 5.6.11 and Satz 5.6.12].

Definition 3.2. Let A and B be C∗-algebras over X (or, more generally,C∗-algebras with a map O(X) → I(A)). A Kasparov cycle (ϕ,HB , F, γ) or(ϕ,HB , F ) for KK∗(A,B) is called X-equivariant if

ϕ(A(U)

)· HB ⊆ HB ·B(U) for all U ∈ O(X).

Let KK∗(X ;A,B) be the group of homotopy classes of such X-equivariantKasparov cycles for KK∗(A,B); a homotopy is an X-equivariant Kasparovcycle for KK∗(A,C([0, 1]) ⊗ B), where we view C([0, 1]) ⊗ B as a C∗-algebraover X in the usual way (compare §2.4).

The subsetHB ·B(U) ⊆ HB is a closed linear subspace by the Cohen-HewittFactorization Theorem.

If X is Hausdorff, then the extra condition in Definition 3.2 is equivalentto C0(X)-linearity of ϕ (compare Proposition 2.11). Thus the above definitionof KK∗(X ;A,B) agrees with the more familiar definition of RKK∗(X ;A,B)in [9].

If X = ⋆ is the one-point space, the X-equivariance condition is empty andwe get the plain Kasparov theory KK∗(⋆;A,B) = KK∗(A,B).

The same arguments as usual show that KK∗(X ;A,B) remains unchangedif we strengthen the conditions for Kasparov cycles by requiring F = F ∗ andF 2 = 1.

3.2. Basic properties. The Kasparov theory defined above has all the prop-erties that we can expect from a bivariant K-theory.

(1) The groups KK∗(X ;A,B) define a bifunctor from C∗sep(X) to thecategory of Z/2-graded abelian groups, contravariant in the first andcovariant in the second variable.

(2) There is a natural, associative Kasparov composition product

KKi(X ;A,B)×KKj(X ;B,C)→ KKi+j(X ;A,C)

if A,B,C are C∗-algebras over X .Furthermore, there is a natural exterior product

KKi(X ;A,B)×KKj(Y ;C,D)→ KKi+j(X × Y ;A⊗ C,B ⊗D)

for two spacesX and Y and C∗-algebrasA, B overX and C, D over Y .The existence and properties of the Kasparov composition product

and the exterior product are verified in a more general context in [4,§3.2].



Definition 3.3. Let KK(X) be the category whose objects are the separableC∗-algebras over X and whose morphism sets are KK0(X ;A,B).

(3) The zero C∗-algebra acts as a zero object in KK(X), that is,

KK∗(X ; 0, A) = 0 = KK∗(X ;A, 0) for all A ∈∈ KK(X).

(4) The C0-direct sum of a sequence of C∗-algebras behaves like a coprod-uct, that is,

KK∗

(X ;⊕

n∈N

An, B

)∼=∏

n∈N

KK∗(X ;An, B)

if An, B ∈∈ KK(X) for all n ∈ N.(5) The direct sum A ⊕ B of two separable C∗-algebras A and B over X

is a direct product in KK(X), that is,

KK∗(X ;D,A⊕B) ∼= KK∗(X ;D,A)⊕KK∗(X ;D,B)

for all D ∈∈ KK(X) (see [4, Lem. 3.1.9]).

Properties (3)–(5) are summarized as follows:

Proposition 3.4. The category KK(X) is additive and has countable coprod-ucts.

(6) The exterior product is compatible with the Kasparov product, C0-direct sums, and addition, that is, it defines a countably additive bi-functor

⊗ : KK(X)⊗ KK(Y )→ KK(X × Y ).

This operation is evidently associative.(7) In particular, KK(X) is tensored over KK(⋆) ∼= KK, that is, ⊗ provides

an associative bifunctor

⊗ : KK(X)⊗ KK→ KK(X).

(8) The bifunctor (A,B) 7→ KK∗(X ;A,B) satisfies Bott periodicity, ho-motopy invariance, and C∗-stability in each variable. This follows fromthe corresponding properties of KK using the tensor structure in (7).

For instance, the Bott periodicity isomorphism C0(R2) ∼= C in KK

yields A⊗ C0(R2) ∼= A⊗ C ∼= A in KK(X) for all A ∈∈ KK(X).(9) The functor f∗ : C∗alg(X)→ C∗alg(Y ) for a continuous map f : X → Y

descends to a functor

f∗ : KK(X)→ KK(Y ).

In particular, this covers the extension functors iYX for a subspace X ⊆Y .

(10) The restriction functor rYX for Y ∈ LC(X) also descends to a functor

rYX : KK(X)→ KK(Y ).



Definition 3.5 (see [4, Def. 5.6.6]). A diagram I → E → Q in C∗alg(X) isan extension if, for all U ∈ O(X), the diagrams I(U) → E(U) → Q(U) areextensions of C∗-algebras. We write I E ։ Q to denote extensions.

An extension is called split if it splits by anX-equivariant ∗-homomorphism.An extension is called semi-split if there is a completely positive, contractive

section Q → E that is X-equivariant, that is, it restricts to sections Q(U) →E(U) for all U ∈ O(X).

If I E ։ Q is an extension of C∗-algebras overX , then we get C∗-algebraextensions I(Y ) E(Y ) ։ Q(Y ) for all locally closed subsets Y ⊆ X . If theoriginal extension is semi-split, so are the extensions I(Y ) E(Y ) ։ Q(Y )for Y ∈ LC(X). Even more, the functor rYX : C∗alg(X) → C∗alg(Y ) mapsextensions in C∗alg(X) to extensions in C∗alg(Y ), and similarly for split andsemi-split extensions.

Theorem 3.6. Let I E ։ Q be a semi-split extension in C∗sep(X) andlet B be a separable C∗-algebra over X. There are six-term exact sequences

KK0(X ;Q,B) // KK0(X ;E,B) // KK0(X ; I, B)

∂

KK1(X ; I, B)

∂

OO

KK1(X ;E,B)oo KK1(X ;Q,B)oo

and

KK0(X ;B, I) // KK0(X ;B,E) // KK0(X ;B,Q)

∂

KK1(X ;B,Q)

∂

OO

KK1(X ;B,E)oo KK1(X ;B, I),oo

where the horizontal maps in both exact sequences are induced by the givenmaps I → E → Q, and the vertical maps are, up to signs, Kasparov productswith the class of our semi-split extension in KK1(Q, I).

Furthermore, extensions with an X-equivariant completely positive sectionare semi-split.

Proof. The long exact sequences for semi-split extensions follow from [4,Satz 3.3.10] or from [4, Kor. 5.6.13].

The last statement of the theorem plays a technical role in the proof ofProposition 4.10. We have to replace an X-equivariant completely positivesection s : Q→ E by another section that is an X-equivarant completely pos-itive contraction. Without X-equivariance, this is done in [6, Rem. 2.5]. Weclaim that the constructions during the proof yield X-equivariant maps if westart with X-equivariant maps.

Let Q+ and E+ be obtained by adjoining units to Q and E. Let (un)n∈N

be an approximate unit in Q and let vn := sup(1, s(un)) in E+. Since vn ≥ 1,vn is invertible. The maps

sn : Q+ → E+, q 7→ v−1/2

n s(u1/2n qu

1/2n )v−

1/2n



considered in [6] are completely positive and contractive because sn(1) ≤ 1.They are X-equivariant if s is X-equivariant. Since vn lifts 1 ∈ Q+, the

maps (sn) lift the maps q 7→ u1/2n qu

1/2n , which converge pointwise to the identity

map.It remains to show that the space of maps Q+ → Q+ that lift to an

X-equivariant unital completely positive map Q+ → E+ is closed in thetopology of pointwise norm convergence. Without X-equivariance, this is [3,Thm. 6]. Its proof is based on the following construction. Let ϕ, ψ : Q+ E+

be two unital completely positive maps and let (em)m∈N be a quasi-central ap-proximate unit for I in Q+. Then Arveson uses the unital completely positivemaps

q 7→ e1/2m ϕ(q)e

1/2m + (1 − em)

1/2ψ(q)(1 − em)1/2.

Clearly, this map is X-equivariant if ϕ and ψ are X-equivariant. Hence theargument in [3] produces a Cauchy sequence ofX-equivariant unital completelypositive maps sn : Q+ → E+ lifting sn. Its limit is an X-equivariant unitalcompletely positive section Q+ → E+.

Theorem 3.7. The canonical functor C∗sep(X) → KK(X) is the universalsplit-exact C∗-stable (homotopy) functor.

Proof. This follows from [4, Satz 3.5.10], compare also [4, Kor. 5.6.13]. Thehomotopy invariance assumption is redundant because, by a deep theoremof Nigel Higson, a split-exact, C∗-stable functor is automatically homotopyinvariant. This holds for C∗sep itself and is inherited by C∗sep(X) because ofthe tensor product operation C∗sep(X)× C∗sep→ C∗sep(X).

3.3. Triangulated category structure. We are going to turn KK(X) intoa triangulated category as in [14]. We have already remarked that KK(X) isadditive. The suspension functor is Σ(A) := C0(R, A) = C0(R) ⊗ A. Thisfunctor is an automorphism (up to natural isomorphisms) by Bott periodicity.

The mapping cone triangle

(3.8) Aϕ // B

~~

Cϕ

``AAAAAA

of a morphism ϕ : A→ B in C∗sep(X) is defined as in [14] and is a diagram inKK(X). The circled arrow from B to Cϕ means a ∗-homomorphism Σ(B) →Cϕ. A triangle in KK(X) is called exact if it is isomorphic in KK(X) to themapping cone triangle of some morphism in C∗sep(X).

As in [14], there is an equivalent description of the exact triangles usingsemi-split extensions in C∗sep(X). An extension

(3.9) Ii

Ep։ Q



gives rise to a commuting diagram

ΣQ Ii //

Ep // Q

ΣQ // Cp // Ep // Q.

Definition 3.10. We call the extension admissible if the map I → Cp isinvertible in KK(X).

The proof of the Excision Theorem 3.6 shows that this is the case if theextension is semi-split; but there are more admissible extensions than semi-split extensions. If the extension is admissible, then there is a unique mapΣQ → I so that the top row becomes isomorphic to the bottom row as atriangle in KK(X). Thus any admissible extension in C∗sep(X) yields an exacttriangle ΣQ→ I → E → Q, called extension triangle.

Conversely, if ϕ : A→ B is a morphism in C∗sep(X), then its mapping conetriangle is isomorphic in KK(X) to the extension triangle for the canonicallysemi-split extension Cϕ Zϕ ։ B, where Zϕ denotes the mapping cylinderof ϕ, which is homotopy equivalent to A. The above arguments work exactlyas in the case of undecorated Kasparov theory discussed in [14].

As a result, a triangle in KK(X) is isomorphic to a mapping cone triangleof some morphism in C∗sep(X) if and only if it is isomorphic to the extensiontriangle of some semi-split extension in C∗sep(X).

Proposition 3.11. The category KK(X) with the suspension automorphismand extension triangles specified above is a triangulated category.

Proof. Most of the axioms amount to well-known properties of mapping conesand mapping cylinders, which are proven by translating corresponding argu-ments for the stable homotopy category of spaces, see [14].

The only axiom that requires a new argument in our case is (TR1), whichasserts that any morphism in KK(X) is part of some exact triangle. Theargument in [14] uses the description of Kasparov theory via the universalalgebra qA by Joachim Cuntz. This approach can be made to work in KK(X),but it is rather inflexible because the primitive ideal space of qA is hard tocontrol.

The following argument, which is inspired by [4], also applies to interestingsubcategories of KK(X) like the subcategory of nuclear C∗-algebras over X,which is studied in §5. Hence this is a triangulated category as well.

Let f ∈ KK0(X ;A,B). We identify KK0(X ;A,B) ∼= KK1(X ;A,ΣB).Represent the image of f in KK1(X ;A,ΣB) by a cycle (ϕ,H, F ). Addinga degenerate cycle, if necessary, we can achieve that the map Φ: A ∋ a 7→F ∗ϕ(a)F mod K(H) is an injection from A into the Calkin algebra B(H)/K(H)of H and that H is full, so that K(H) is KK(X)-equivalent to ΣB. The prop-erties of a Kasparov cycle mean that Φ is the Busby invariant of a semi-splitextension K(H) E ։ A of C∗-algebras over X . The composition product



of the map ΣA → K(H) in the associated extension triangle and the canon-ical KK(X)-equivalence K(H) ≃ ΣB is the suspension of f ∈ KK0(X ;A,B).Hence we can embed f in an exact triangle.

3.4. Adjointness relations.

Proposition 3.12. Let X be a topological space and let Y ∈ LC(X).If Y ⊆ X is open, then we have natural isomorphisms

KK∗(X ; iXY (A), B) ∼= KK∗(Y ;A, rYX(B)

)

for all A ∈∈ KK(Y ), B ∈∈ KK(X), that is, iXY is left adjoint to rYX as functorsKK(Y )↔ KK(X).

If Y ⊆ X is closed, then we have natural isomorphisms

KK∗(Y ; rYX(A), B) ∼= KK∗(X ;A, iXY (B)

)

for all A ∈∈ KK(X), B ∈∈ KK(Y ), that is, iXY is right adjoint to rYX as functorsKK(Y )↔ KK(X).

Proof. Since both iXY and rYX descend to functors between KK(X) and KK(Y ),this follows from the adjointness on the level of C∗alg(X) and C∗alg(Y ) inLemma 2.20; an analogous assertion for induction and restriction functorsfor group actions on C∗-algebras is proven in [14, §3.2]. The point of theargument is that an adjointness relation is equivalent to the existence of certainnatural transformations called unit and counit of the adjunction, subject tosome conditions (see [12]). These natural transformations already exist on thelevel of ∗-homomorphisms, which induce morphisms in KK(X) or KK(Y ). Thenecessary relations for unit and counit of adjunction hold in KK(. . . ) becausethey already hold in C∗alg(. . . ). The unit and counit are natural in KK(. . . )and not just in C∗alg(. . . ) because of the uniqueness part of the universalproperty of KK.

Proposition 3.13. Let X be a topological space and let x ∈ X. Then

KK∗(X ;A, ix(B)

) ∼= KK∗(A(x), B)

for all A ∈∈ C∗sep(X), B ∈∈ C∗sep. That is, the functor ix : KK→ KK(X) is

right adjoint to the functor A 7→ A(x). Moreover,

KK∗(X ; ix(A), B) ∼= KK∗

(A,

⋂

U∈Ux

B(U)

)

for all A ∈∈ C∗sep, B ∈∈ C∗sep(X), where Ux denotes the open neighborhoodfilter of x in X, That is, the functor ix : KK → KK(X) is left adjoint to thefunctor B 7→ ⋂

U∈UxB(U). If x has a minimal open neighborhood Ux, then

KK∗(X ; ix(A), B) ∼= KK∗(A,B(Ux)

),

Proof. This follows from Lemma 2.22 in the same way as Proposition 3.12. No-tice that B 7→ ⋂

U∈UxB(U) commutes with C∗-stabilization and maps (semi)-

split extensions in C∗alg(X) again to (semi)-split extensions in C∗alg; therefore,it descends to a functor KK(X)→ KK.



4. The bootstrap class

Throughout this section, X denotes a finite and sober topological space.Finiteness is crucial here. First we construct a canonical filtration on anyC∗-algebra over X . We use this to study the analogue of the bootstrap classin KK(X). Along the way, we also introduce the larger category of localC∗-algebras over X . Roughly speaking, locality means that all the canoni-cal C∗-algebra extensions that we get from C∗-algebras over X are admissible.Objects in the X-equivariant bootstrap category have the additional propertythat their fibers belong to the usual bootstrap category.

4.1. The canonical filtration. We recursively construct a canonical filtra-tion

∅ = F0X ⊂ F1X ⊂ · · · ⊂ FℓX = X

of X by open subsets FjX , such that the differences

Xj := FjX \ Fj−1X

are discrete for all j = 1, . . . , ℓ. In each step, we let Xj be the subset of all openpoints in X \ Fj−1X—so that Xj is discrete—and put FjX = Fj−1X ∪ Xj.Equivalently, Xj consists of all points of X \ Fj−1X that are maximal for thespecialization preorder≺. SinceX is finite, Xj is nonempty unless Fj−1X = X ,and our recursion reaches X after finitely many steps.

Definition 4.1. The length ℓ of X is the length of the longest chain x1 ≺x2 ≺ · · · ≺ xℓ in X .

We assume X finite to ensure that the above filtration can be constructed.It is easy to extend our arguments to Alexandrov spaces of finite length; theonly difference is that the discrete spaces Xj may be infinite in this case, sothat we need infinite direct sums in some places, forcing us in Proposition 4.7to drop (2) and replace the words “triangulated” by “localizing” in the lastsentence. It should be possible to treat Alexandrov spaces of infinite length ina similar way. Since such techniques cannot work for non-Alexandrov spaces,anyway, we do not pursue these generalizations here.

Definition 4.2. We shall use the functors

PY := iXY rYX : C∗alg(X)→ C∗alg(X)

for Y ∈ LC(X). Thus (PY A)(Z) ∼= A(Y ∩ Z) for all Z ∈ LC(X).

If Y ∈ LC(X), U ∈ O(Y ), then we get an extension

(4.3) PU (A) PY (A) ։ PY \U (A)

in C∗alg(X) because of the extensions A(Z ∩U) A(Z ∩ Y ) ։ A(Z ∩ Y \U)for all Z ∈ LC(X).

Let A be a C∗-algebra over X . We equip A with the canonical increasingfiltration by the ideals

FjA := PFjX(A), j = 0, . . . , ℓ,



so that

(4.4) FjA(Y ) = A(Y ∩ FjX) = A(Y ) ∩A(FjX) for all Y ∈ LC(X).

Equation (4.3) shows that the subquotients of this filtration are(4.5)

FjA / Fj−1A ∼= PFjX\Fj−1X(A) = PXj (A) ∼=⊕

x∈Xj

Px(A) =⊕

x∈Xj

ix(A(x)

).

Here ix = iXx for x ∈ X denotes the extension functor from the subset x ⊆ X :

ix : KK∼=−→ KK(x) ix−→ KK(X), (ixB)(Y ) =

B if x ∈ Y ,

0 if x /∈ Y .

Example 4.6. Consider the space X = 1, 2 with the nondiscrete topologydescribed in Example 2.29. Here

F0X = ∅, F1X = 1, F2X = 1, 2 = X, X1 = 1, X2 = 2.The filtration FjA on a C∗-algebra overX has one nontrivial layer F1A becauseF0A = 0 and F2A = A. Recall that C∗-algebras over X correspond toextensions of C∗-algebras. For a C∗-algebra extension I A ։ A/I, the firstfiltration layer is simply the extension I I ։ 0, so that the quotient A/F1Ais the extension 0 A/I ։ A/I. Our filtration decomposes I A ։ A/Iinto an extension of C∗-algebra extensions as follows:

(I I ։ 0) (I A ։ A/I) ։ (0 A/I ։ A/I).

Proposition 4.7. The following are equivalent for a separable C∗-algebra Aover X:

(1) The extensions Fj−1A FjA ։ FjA/Fj−1A in C∗sep(X) are admis-sible for j = 1, . . . , ℓ.

(2) A ∈∈ KK(X) belongs to the triangulated subcategory of KK(X) gener-ated by objects of the form ix(B) with x ∈ X, B ∈∈ KK.

(3) A ∈∈ KK(X) belongs to the localizing subcategory of KK(X) generatedby objects of the form ix(B) with x ∈ X, B ∈∈ KK.

(4) For any Y ∈ LC(X), U ∈ O(Y ), the extension

PU (A) PY (A) ։ PY \U (A)

in C∗sep(X) described above is admissible.

Furthermore, if A satisfies these conditions, then it already belongs to the tri-angulated subcategory of KK(X) generated by ix

(A(x)

)for x ∈ X.

Recall that the localizing subcategory generated by a family of objects inKK(X) is the smallest subcategory that contains the given objects and is tri-angulated and closed under countable direct sums.

Proof. (2)=⇒(3) and (4)=⇒(1) are trivial. We will prove (1)=⇒(2) and(3)=⇒(4).



(1)=⇒(2): Since the extensions Fj−1A FjA ։ FjA/Fj−1A are admissible,they yield extension triangles in KK(X). Thus FjA belongs to thetriangulated subcategory of KK(X) generated by Fj−1A and FjA /Fj−1A. Since F0A = 0, induction on j and (4.5) show that FjA belongsto the triangulated subcategory generated by ixA(x) with x ∈ FjX .Thus A = FℓA belongs to the triangulated subcategory of KK(X)generated by ix

(A(x)

)for x ∈ X . This also yields the last statement

in the proposition.(3)=⇒(4): It is clear that (4) holds for objects of the form ix(B) because

at least one of the three objects PU ix(B), PY ix(B), or PY \U ix(B)vanishes. The property (4) is inherited by (countable) direct sums,suspensions, and mapping cones. To prove the latter, we use the def-inition of admissibility as an isomorphism statement in KK(X) andthe Five Lemma in triangulated categories. Hence (4) holds for allobjects of the localizing subcategory generated by ix(B) for x ∈ X ,B ∈∈ KK.

Definition 4.8. Let KK(X)loc ⊆ KK(X) be the full subcategory of all objectsthat satisfy the equivalent conditions of Proposition 4.7.

The functor f∗ : KK(X)→ KK(Y ) for a continuous map f : X → Y restrictsto a functor KK(X)loc → KK(Y )loc because f∗ iXx = iYx and f∗ is an exactfunctor. Similarly, the restriction functor rYX : KK(X) → KK(Y ) for a locallyclosed subset Y ⊆ X maps KK(X)loc to KK(Y )loc because it is exact and rYX iXxis iYx for x ∈ Y and 0 otherwise.

Proposition 4.9. Let X be a finite topological space. Let A,B ∈ KK(X)loc

and let f ∈ KK∗(X ;A,B). If f(x) ∈ KK∗(A(x), B(x)

)is invertible for all

x ∈ X, then f is invertible in KK(X). In particular, if A(x) ∼= 0 in KK for allx ∈ X, then A ∼= 0 in KK(X).

Proof. The second assertion follows immediately from the last sentence inProposition 4.7. It implies the first one by a well-known trick: embed α inan exact triangle by axiom (TR1) of a triangulated category, and use thelong exact sequence to relate invertibility of α to the vanishing of its mappingcone.

Proposition 4.10. Suppose that the C∗-algebra extensions

A(Ux \ x) A(Ux) ։ A(x)

are semi-split for all x ∈ X. Then A ∈∈ KK(X)loc. In particular, this appliesif the underlying C∗-algebra of A ∈∈ KK(X) is nuclear.

Proof. We claim that the extensions in Proposition 4.7.(1) are semi-split asextensions of C∗-algebras overX , hence admissible in KK(X). For this, we needa completely positive section A(Xj)→ A(FjX) that is X-equivariant, that is,restricts to maps A(Xj ∩ V ) → A(FjX ∩ V ) for all V ∈ O(FjX). We takethe sum of the completely positive sections for the extensions A(Ux \ x)



A(Ux) ։ A(x) for x ∈ Xj . This map has the required property becauseany open subset containing x also contains Ux; it is irrelevant whether ornot this section is contractive by the last sentence in Theorem 3.6. If A isnuclear, so are the ideals A(Ux) and their quotients A(x) for x ∈ X . Thus theabove extensions have completely positive sections by the Choi-Effros LiftingTheorem (see [5]).

It is not clear whether the mere admissibility in C∗sep of the extensions

A(Ux \ x) A(Ux) ։ A(x)

suffices to conclude that A ∈∈ KK(X)loc. This condition is certainly necessary.

The constructions above yield spectral sequences as in [19]. These may beuseful for spaces of length 1, where they degenerate to a short exact sequence.We only comment on this very briefly.

Let A ∈∈ KK(X)loc. The admissible extensions Fj−1A FjA ։ FjA /Fj−1A for j = 1, . . . , ℓ produce exact triangles in KK(X). A homological orcohomological functor such as KK(X ;D, ) or KK(X ; , D) maps these exacttriangles to a sequence of exact chain complexes. These can be arranged inan exact couple, which generates a spectral sequence (see [11]). This spectralsequence could, in principle, be used to compute KK∗(X ;A,B) in terms of

KK∗(X ; FjA / Fj−1A,B) ∼=∏

x∈Xj

KK∗(X ; ixA(x), B)

∼=∏

x∈Xj

KK∗(A(x), B(Ux)

),

where we have used Proposition 3.13. These groups comprise the E1-termsof the spectral sequence that we get from our exact couple for the functorKK(X ; , B).

For instance, consider again the situation of Example 4.6. Let I⊳A and J⊳Bbe C∗-algebras over X , corresponding to C∗-algebra extensions I A ։ A/Iand J B ։ B/J . The above spectral sequence degenerates to a long exactsequence

KK0(A/I,B) // KK0(X ; I ⊳ A, J ⊳ B) // KK0(I, J)

δ

KK1(I, J)

δ

OO

KK1(X ; I ⊳ A, J ⊳ B)oo KK1(A/I,B).oo

The boundary map is the diagonal map in the following commuting diagram:

KK0(I, J)

//

δ

((QQQQQQQQQQQQKK0(I, B)

KK1(A/I, J) // KK1(A/I,B).



We can rewrite the long exact sequence above as an extension:

coker δ KK∗(X ; I ⊳ A, J ⊳ B) ։ ker δ.

But we lack a description of ker δ and coker δ as Hom- and Ext-groups. There-fore, the Universal Coefficient Theorem of Alexander Bonkat [4] seems moreattractive.

4.2. The bootstrap class. The bootstrap class B in KK is the localizing sub-category generated by the single object C, that is, it is the smallest class of sep-arable C∗-algebras that contains C and is closed under KK-equivalence, count-able direct sums, suspensions, and the formation of mapping cones (see [16]).

A localizing subcategory of KK(X) or KK is automatically closed undervarious other constructions, as explained in [14]. This includes admissibleextensions, admissible inductive limits (the appropriate notion of admissibilityis explained in [14]), and crossed products by Z and R and, more generally, byactions of torsion-free amenable groups.

The latter result uses the reformulation of the (strong) Baum-Connes prop-erty for such groups in [14]. This reformulation asserts that C with the trivialrepresentation of an amenable group G belongs to the localizing subcategoryof KK(G) generated by C0(G). Carrying this over to KK(X), we concludethat A ⋊ G for A ∈∈ KK(X) belongs to the localizing subcategory of KK(X)generated by

(A⊗ C0(G)

)⋊G, which is Morita-Riefel equivalent to A.

The following definition provides an analogue B(X) ⊆ KK(X) of the boot-strap class B ⊆ KK for a finite topological space X :

Definition 4.11. Let B(X) be the localizing subcategory of KK(X) that isgenerated by ix(C) for x ∈ X .

Notice that ix(C) | x ∈ X lists all possible ways to turn C into aC∗-algebra over X .

Proposition 4.12. Let X be a finite topological space and let A ∈∈ KK(X).The following conditions are equivalent:

(1) A ∈∈ B(X);(2) A ∈∈ KK(X)loc and A(x) ∈∈ B for all x ∈ X;(3) the extensions Fj−1A FjA ։ FjA / Fj−1A are admissible for j =

1, . . . , ℓ, and A(x) ∈∈ B for all x ∈ X;

In addition, in this case A(Y ) ∈∈ B for all Y ∈ LC(X).

Proof. The equivalence of (2) and (3) is already contained in Proposition 4.7.Using the last sentence of Proposition 4.7, we also get the implication (3)=⇒(1)because ix is exact and commutes with direct sums. The only assertion thatis not yet contained in Proposition 4.7 is that A ∈∈ B(X) implies A(Y ) ∈∈ Bfor all Y ∈ LC(X). The reason is that the functor KK(X)→ KK, A 7→ A(Y ),is exact, preserves countable direct sums, and maps the generators iy(C) fory ∈ X to either 0 or C and hence into B.



Corollary 4.13. If the underlying C∗-algebra of A is nuclear, then A ∈∈ B(X)if and only if A(x) ∈∈ B for all x ∈ X.

Proof. Combine Propositions 4.10 and 4.12.

Example 4.14. View a separable nuclear C∗-algebra A with only finitely manyideals as a C∗-algebra over Prim(A). Example 2.16 and Corollary 4.13 showthat A belongs to B(PrimA) if and only if all its simple subquotients belongto the usual bootstrap class in KK.

Proposition 4.15. Let X be a finite topological space. Let A,B ∈∈ B(X) andlet f ∈ KK∗(X ;A,B). If f induces invertible maps K∗

(A(x)

)→ K∗

(B(x)

)

for all x ∈ X, then f is invertible in KK(X). In particular, if K∗(A(x)

)= 0

for all x ∈ X, then A ∼= 0 in KK(X).

Proof. As in the proof of Proposition 4.9, it suffices to show the second asser-tion. Since A(x) ∈∈ B for all x ∈ X , vanishing of K∗

(A(x)

)implies vanishing

of A(x) in KK, so that Proposition 4.9 yields the assertion.

4.3. Complementary subcategories. It is often useful to replace a givenobject of KK(X) by one in the bootstrap class or KK(X)loc that is as close tothe original as possible. This is achieved by localization functors

LB : KK(X)→ B(X), L : KK(X)→ KK(X)loc

that are right adjoint to the embeddings of these subcategories. That is, wewant KK∗

(X ;A,LB(B)

) ∼= KK∗(X ;A,B) for all A ∈∈ B(X), B ∈ KK(X)and similarly for L. These functors come with natural transformations LB ⇒L ⇒ id, and the defining property is equivalent to L(A)x → Ax being aKK-equivalence and K∗(LB(A)x) → K∗(Ax) being invertible for all x ∈ X ,respectively.

The functors L and LB exist because our two subcategories belong to com-plementary pairs of localizing subcategories in the notation of [14]. The exis-tence of this complementary pair is straightforward to prove using the tech-niques of [13].

Definition 4.16. Let KK(X)⊣loc be the class of all A ∈∈ KK(X) for which A(x)is KK-equivalent to 0 for all x ∈ X . Let B(X)⊣ be the class of all A ∈∈ KK(X)with K∗

(A(x)

)= 0.

Theorem 4.17. The pair of subcategories (B(X),B(X)⊣) is complementary.So is the pair (KK(X)loc,KK(X)⊣loc).

Proof. We first prove the assertion for KK(X)loc. Consider the exact functor

F : KK(X)→∏

x∈X

KK, A 7→ (Ax)x∈X .

Let I be the kernel of F on morphisms. Since F is an exact functor thatcommutes with countable direct sums, I is a stable homological ideal thatis compatible with direct sums (see [15, 13]). The kernel of F on objects is



exactly KK(X)⊣. Proposition 3.13 shows that the functor F has a left adjoint,namely, the functor

F⊢((Ax)x∈X

):=⊕

x∈X

ix(Ax).

Therefore, the ideal I has enough projective objects by [15, Prop. 3.37]; fur-thermore, the projective objects are retracts of direct sums of objects of theform ix(Ax). Hence the localizing subcategory generated by the I-projectiveobjects is KK(X)loc by Proposition 4.7. Finally, [13, Thm. 4.6] shows that thepair of subcategories (KK(X)loc,KK(X)⊣loc) is complementary.

The argument for the bootstrap category is almost literally the same, but us-

ing the stable homological functor K∗F : KK(X)→∏x∈X AbZ/2

c instead of F ,

where AbZ/2c denotes the category of countable Z/2-graded abelian groups. The

adjoint of K∗ F is defined on families of countable free abelian groups, whichis enough to conclude that ker(K∗ F ) has enough projective objects. Thistime, the projective objects generate the category B(X), and the kernel of Fon objects is B(X)⊣. Hence [13, Thm. 4.6] shows that the pair of subcategories(B(X),B(X)⊣) is complementary.

Lemma 4.18. The following are equivalent for A ∈∈ KK(X):

(1) K∗(A(x)

)= 0 for all x ∈ X, that is, A ∈∈ B(X)⊣;

(2) K∗(A(Y )

)= 0 for all Y ∈ LC(X);

(3) K∗(A(U)

)= 0 for all U ∈ O(X).

Proof. It is clear that (2) implies both (1) and (3). Conversely, (3) implies (2):write Y ∈ LC(X) as U \ V with U, V ∈ O(X), V ⊆ U , and use the K-theorylong exact sequence for the extension A(U) A(V ) ։ A(Y ). It remains tocheck that (1) implies (2).

We prove by induction on j that (1) implies K∗(A(Y )

)= 0 for all Y ∈

LC(FjX). This is trivial for j = 0. If Y ⊆ Fj+1X , then K∗(A(Y ∩ FjX)

)=

0 by the induction assumption. The K-theory long exact sequence for theextension

A(Y ∩ FjX) A(Y ) ։⊕

x∈Xj+1∩Y

A(x)

yields K∗(A(Y )

)= 0 as claimed.

We can also apply the machinery of [15, 13] to the ideal I to generatea spectral sequence that computes KK∗(X ;A,B). This spectral sequence ismore useful than the one from the canonical filtration because its second pageinvolves derived functors. But this spectral sequence rarely degenerates to anexact sequence.

4.4. A definition for infinite spaces. The ideas in §4.3 suggest a definitionof the bootstrap class for infinite spaces.

Definition 4.19. Let X be a topological space. Let B(X)⊣ ⊆ KK(X) consistof all separable C∗-algebras over X with K∗

(A(U)

)= 0 for all U ∈ O(X).



Lemma 4.18 shows that this agrees with our previous definition for finite X .Furthermore, the same argument as in the proof of Lemma 4.18 yields A ∈∈B(X)⊣ if and only if K∗

(A(Y )

)= 0 for all Y ∈ LC(X). The first condition in

Lemma 4.18 has no analogue because of Example 2.17.It is clear from the definition that B(X)⊣ is a localizing subcategory of

KK(X).

Definition 4.20. Let X be a topological space. We let B(X) be the localiza-tion of KK(X) at B(X)⊣.

For finite X , we have seen that B(X)⊣ is part of a complementary pair oflocalizing subcategories, with partner B(X). This shows that the localizationof KK(X) at B(X)⊣ is canonically equivalent to B(X). For infinite X , it isunclear whether B(X)⊣ is part of a complementary pair. If it is, the partnermust be

D := A ∈∈ KK(X) | KK∗(X ;A,B) = 0 for all B ∈∈ B(X)⊣.Since B ∈∈ B(X)⊣ implies nothing about the K-theory of

⋂U∈Ux

B(U), ingeneral, Proposition 3.13 shows that ixC does not belong to D in general.

If X is Hausdorff, then C0(U) ∈∈ B(X)⊣ for all U ∈ O(X). Nevertheless,it is not clear whether (D,B(X)⊣) is complementary.

5. Making the fibers simple

Definition 5.1. A C∗-algebra (A,ψ) overX is called tight if ψ : Prim(A)→ Xis a homeomorphism.

Tightness implies that the fibers Ax = A(x) for x ∈ X are simple C∗-alge-bras. But the converse does not hold: the fibers are simple if and only if themap ψ : Prim(A)→ X is bijective.

To equip KK(X) with a triangulated category structure, we must drop thetightness assumption because it is usually destroyed when we construct cylin-ders, mapping cones, or extensions of C∗-algebras over X . Nevertheless, weshow below that we may reinstall tightness by passing to a KK(X)-equivalentobject, at least in the nuclear case.

The special case where the space X in question has only one point is alreadyknown:

Theorem 5.2 ([18, Prop. 8.4.5]). Any separable nuclear C∗-algebra is KK-equivalent to a C∗-algebra that is separable, nuclear, purely infinite, C∗-stableand simple.

Stability is not part of the assertion in [18], but can be achieved by tensoringwith the compact operators, without destroying the other properties. The maindifficulty is to achieve simplicity. We are going to generalize this theorem asfollows:

Theorem 5.3. Let X be a finite topological space. Any separable nuclearC∗-algebra over X is KK(X)-equivalent to a C∗-algebra over X that is tight,separable, nuclear, purely infinite, and C∗-stable.



For the zero C∗-algebra, viewed as a C∗-algebra over X , this reproves theknown statement that there is a separable, nuclear, purely infinite, and stableC∗-algebra with spectrum X for any finite topological space X .

Proof. Since A is separable and nuclear, so are the subquotients Ax. HenceTheorem 5.2 provides simple, separable, nuclear, stable, purely infinite C∗-algebras Bx and KK-equivalences fx ∈ KK0(Ax, Bx) for all x ∈ X .

We use the canonical filtration FjX of X and the resulting filtration FjAintroduced in §4.1, see (4.4). The subquotients

A0j := FjA / Fj−1A

of the filtration are described in (4.5) in terms of the subquotients Ax forx ∈ X .

We will recursively construct a sequence Bj of C∗-algebras over X that aresupported on FjX and KK(X)-equivalent to FjA for j = 0, . . . , ℓ, such thatFjBk = Bj for k ≥ j and each Bj is tight over FjX , separable, nuclear, purelyinfinite, and stable. The last object Bℓ in this series is KK(X)-equivalent toFℓA = A and has all the required properties. Since F0X = ∅, the recursionmust begin with B0 = A0 = 0. We assume that Bj has been constructed.Let

B0j+1 :=

⊕

x∈Xj+1

ix(Bx).

We will construct Bj+1 as an extension of Bj by B0j+1. This ensures that the

fibers of Bj are Bx for x ∈ FjX and 0 for x ∈ X \ FjX .First we construct, for each x ∈ Xj+1, a suitable extension of Bx by Bj . Let

Ux ⊆ Fj+1X be the minimal open subset containing x and let U ′x := Ux \ x.Since Xj+1 is discrete, U ′x is an open subset of FjX . The extension

A(U ′x) A(Ux) ։ Ax

is semi-split and thus provides a class δAx in KK1

(Ax, A(U ′x)

)because Ax is

nuclear. Since Bx ≃ Ax, FjA ≃ Bj and FjA(U ′x) = A(U ′x), we can transformthis class to δBx in KK1

(Bx, Bj(U

′x)).

We abbreviate Bjx := Bj(U′x) to simplify our notation. Represent δBx by

an odd Kasparov cycle (H, ϕ, F ), where H is a Hilbert Bjx-module, ϕ : Bx →B(H) is a ∗-homomorphism, and F ∈ B(H) satisfies F 2 = 1, F = F ∗, and[F, ϕ(b)] ∈ K(H) for all b ∈ Bx. Now we apply the familiar correspondencebetween odd KK-elements and C∗-algebra extensions. Let P := 1

2 (1+F ), then

ψ : Bx → B(H) /K(H), b 7→ Pϕ(x)P

is a ∗-homomorphism and hence the Busby invariant of an extension of Bx byK(H). After adding a sufficiently big split extension, that is, a ∗-homomor-phism ψ0 : Bx → B(H′), the map ψ : Bx → B(H)/K(H) becomes injective andthe ideal in K(H) generated by K(H)ψ(Bx)K(H) is all of K(H). We assumethese two extra properties from now on.

We also add to ψ the trivial extension Bj Bj ⊕Bx ։ Bx, whose Busbyinvariant is the zero map. This produces an extension ofBx by K(Bj⊕H) ∼= Bj ;



the last isomorphism holds because Bj is stable, so that Bj ⊕H′ ∼= Bj for anyHilbert Bj-module H′. Since ψ is injective, the extension we get is of theform Bj Ejx ։ Bx. This extension is still semi-split, and its class inKK1(Bx, Bj) is the composite of δBx with the embedding Bjx → Bj. Ourcareful construction ensures that the ideal in Bj generated by Bjψ(Bx)Bj isequal to B(U ′x).

Now we combine these extensions for all x ∈ X by taking their externaldirect sum. This is an extension of

⊕x∈Xj+1

Bx = B0j+1 by the C∗-algebra of

compact operators on the Hilbert Bj-module⊕

x∈Xj+1Bj ∼= Bj , where we used

the stability of Bj once more. Thus we obtain an extension Bj Bj+1 ։

B0j+1. We claim that the primitive ideal space of Bj+1 identifies naturally

with Fj+1X .The extension Bj Bj+1 ։ B0

j+1 decomposes Prim(Bj+1) into an open

subset homeomorphic to Prim(Bj) ∼= FjX and a closed subset homeomorphicto the discrete set Prim(B0

j+1) = Xj+1. This provides a canonical bijection

between Prim(Bj+1) and Fj+1X . We must check that it is a homeomorphism.First let U ⊆ Fj+1X be open in Fj+1X . Then U ∩ FjX is open and con-

tains U ′x for each x ∈ U ∩Xj+1. Our construction ensures that ψ(Bx) ⊆ Bj+1

multiplies Bj into Bjx ⊆ Bj(U ∩ FjX). Hence

Bj(U ∩ FjX) +∑

x∈U∩Xj+1

ψ(Bx)

is an ideal in Bj+1. This shows that U is open in Prim(Bj+1).Now let U ⊆ Fj+1X be open in Prim(Bj+1). Then U ∩ Fj must be open in

FjX ∼= Prim(Bj). Furthermore, if x ∈ U ∩Xj+1, then the subset of Prim(Bj)corresponding to the ideal in Bj generated by Bjψ(Bx)Bj is contained in U .But our construction ensures that this subset is precisely U ′x. Hence

U = (U ∩ FjX) ∪⋃

x∈U∩Xj+1

Ux,

proving that U is open in the topology of Fj+1X . This establishes that ourcanonical map between Prim(Bj+1) and Fj+1X is a homeomorphism. Thuswe may view Bj+1 as a C∗-algebra over X supported in Fj+1X . It is clearfrom our construction that Bj Bj+1 ։ B0

j+1 is an extension of C∗-algebras

over X . Here we view B0j+1 as a C∗-algebra over X in the obvious way, so

that Bx is its fiber over x for x ∈ Xj+1.There is no reason to expect Bj+1 to be stable or purely infinite. But this is

easily repaired by tensoring with K⊗O∞. This does not change Bj and B0j+1,

up to isomorphism, because these are already stable and purely infinite, andit has no effect on the primitive ideal space, nuclearity or separability. Thuswe may achieve that Bj+1 is stable and purely infinite.

By assumption, there is a KK(X)-equivalence fj ∈ KK0(X ; FjA,Bj). Fur-thermore, our construction of B0

j+1 ensures a KK(X)-equivalence f0j+1 between



A0j+1 and B0

j+1. Due to the nuclearity of A, the arguments in §4.1 show that

FjA Fj+1A ։ A0j+1

is a semi-split extension of C∗-algebras over X and hence provides an ex-act triangle in KK(X). The same argument provides an extension trianglefor the extension Bj Bj+1 ։ B0

j+1. Let δAj and δBj be the classes in

KK1(X ;A0j+1,FjA) and KK1(B

0j+1, Bj) associated to these extension; they

appear in the exact triangles described above.Both classes δAj and δBj are, essentially, the sum of the classes δAx and δBx

for x ∈ Xj+1, respectively. More precisely, we have to compose each δAx withthe embedding A(U ′x)→ FjA. Hence the solid square in the diagram

ΣA0j+1

δAj //

Σf0j+1

∼=

FjA //

fj ∼=

Fj+1A //

fj+1 ∼=

A0j+1

f0j+1

∼=

ΣB0

j+1

δBj // Bj // Bj+1 // B0j+1

commutes. By an axiom of triangulated categories, we can find the dottedarrow making the whole diagram commute. The Five Lemma for triangulatedcategories asserts that this arrow is invertible because fj and f0

j+1 are. Thisshows that Bj+1 has all required properties and completes the induction step.

Theorem 5.4. Let X be a finite topological space and let A be a separableC∗-algebra over X. The following are equivalent:

• A ∈∈ KK(X)loc and Ax is KK-equivalent to a nuclear C∗-algebra foreach x ∈ X;• A is KK(X)-equivalent to a C∗-algebra over X that is tight, separable,

nuclear, purely infinite, and C∗-stable.

Proof. The proof of Theorem 5.3 still works under the weaker assumption thatA ∈∈ KK(X)loc and Ax is KK-equivalent to a nuclear C∗-algebra for eachx ∈ X . The converse implication is trivial.

Corollary 5.5. Let X be a finite topological space and let A be a separableC∗-algebra over X. The following are equivalent:

• A ∈∈ B(X);• A is KK(X)-equivalent to a C∗-algebra over X that is tight, separable,

nuclear, purely infinite, C∗-stable, and has fibers Ax in the bootstrapclass B.

Proof. Combine Theorem 5.4 and Proposition 4.12.



By a deep classification result by Eberhard Kirchberg (see [10]), two tight,separable, nuclear, purely infinite, stable C∗-algebras over X are KK(X)-equivalent if and only if they are isomorphic as C∗-algebras over X . There-fore, the representatives found in Theorems 5.3 and 5.4 are unique up toX-equivariant ∗-isomorphism.

Let KK(X)nuc be the subcategory of KK(X) whose objects are the separablenuclear C∗-algebras over X . This is a triangulated category as well becausethe basic constructions like suspensions, mapping cones, and extensions neverleave this subcategory. The subcategory of KK(X) whose objects are the tight,separable, nuclear, purely infinite, stable C∗-algebras over X is equivalent toKK(X)nuc by Theorem 5.3 and hence inherits a triangulated category structure.It has the remarkable feature that isomorphisms in this triangulated categorylift to X-equivariant ∗-isomorphisms.

Recall that a C∗-algebra belongs to B if and only if it is KK-equivalent toa commutative C∗-algebra. This probably remains the case at least for finitespaces X , but the authors do not know how to prove this. For infinite spaces,it is even less clear whether B(X) is equivalent to the KK(X)-category ofcommutative C∗-algebras over X . We only have the following characterization:

Theorem 5.6. A separable C∗-algebra over X belongs to the bootstrap classB(X) if and only if it is KK(X)-equivalent to a C∗-stable, separable C∗-algebraover X of type I.

Proof. Follow the proof of Theorem 5.3, but using stabilizations of commuta-tive C∗-algebras Bx instead of nuclear purely infinite ones. The proof showsthat we can also achieve that the fibers Bx are all of the form C0(Yx)⊗K forsecond countable locally compact spaces Yx.

6. Outlook

We have defined a bootstrap class B(X) ⊆ KK(X) over a finite topologicalspaceX , which is the domain on which we should expect a Universal CoefficientTheorem to compute KK∗(X ;A,B). We have seen that any object of thebootstrap class is KK(X)-equivalent to a tight, purely infinite, stable, nuclear,separable C∗-algebra over X , for which Kirchberg’s classification results apply.

There are several spectral sequences that compute KK∗(X ;A,B), but ap-plications to the classification programme require a short exact sequence. Forsome finite topological spaces, such a short exact sequence is constructed in [16]based on filtrated K-theory, so that filtrated K-theory is a complete invariant.This invariant comprises the K-theory K∗

(A(Y )

)of all locally closed subsets Y

of X together with the action of all natural transformations between them.This is a consequence of a Universal Coefficient Theorem in this case. It isalso shown in [16] that there are finite topological spaces for which filtratedK-theory is not yet a complete invariant. At the moment, it is unclear whetherthere is a general, tractable complete invariant for objects of B(X).



Another issue is to treat infinite topological spaces. A promising approachis to approximate infinite spaces by finite non-Hausdorff spaces associated toopen coverings of the space in question. In good cases, there should be alim←−

1-sequence that relates KK∗(X ;A,B) to Kasparov groups over such finiteapproximations to X , reducing computations from the infinite to the finitecase. Such an exact sequence may be considerably easier for X-equivariantE-theory, where we do not have to worry about completely positive sections.

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Received May 30, 2008; accepted December 18, 2008

Ralf MeyerMathematisches Institut and Courant Research Centre “Higher Order Structures”Georg-August-Universitat GottingenBunsenstraße 3–5, 37073 Gottingen, GermanyE-mail: [email protected]

Ryszard NestKøbenhavns Universitets Institut for Matematiske FagUniversitetsparken 5, 2100 København, DenmarkE-mail: [email protected]




Die universelle unbeschrankte Derivation

Michael Puschnigg


Joachim Cuntz zum 60. Geburtstag

Abstract. We introduce and study the universal unbounded derivation of a Frechet algebrawith fixed dense domain of definition. As an application we construct new examples ofholomorphically closed subalgebras of the reduced C∗-algebra of a nonabelian free group.Every trace on the group ring which is supported in finitely many conjugacy classes extendsto a bounded trace on these algebras.

1. Einleitung

Die Suche nach glatten Unteralgebren von C∗-Algebren ist ein wichtigesProblem der Nichtkommutativen Geometrie. Dabei nennen wir eine dichte Un-teralgebra einer Banachalgebra glatt, falls die Spektren ihrer gemeinsamen Ele-mente in beiden Algebren ubereinstimmen. Typische Beispiele liefern etwa dieUnteralgebra C∞(M) der glatten Funktionen in der C∗-Algebra C(M) aller ste-tigen Funktionen auf einer MannigfaltigkeitM und die Unteralgebra ℓ1(H) derSpurklasse-Operatoren in der C∗-Algebra K(H) aller kompakten Operatorenauf einem Hilbertraum H. In dieser Arbeit benutzen wir eine universelle Kon-struktion, um neue Beispiele glatter Unteralgebren von C∗-Algebren diskreterGruppen zu konstruieren.

Die meisten bekannten Beispiele glatter Unteralgebren von C∗-Algebrenwerden als Abschlusse von Graphen unbeschrankter, dicht definierter Deri-vationen auf der betrachteten Algebra realisiert. Die dabei verwendeten Deri-vationen ergeben sich aus der Geometrie und Analysis der konkreten Situation.Hier beschreiten wir einen anderen Weg und betrachten fur eine FrechetalgebraA die universelle unbeschrankte Derivation dAS mit fixiertem, dichtem Definiti-onsbereich S ⊂ A. Diese existiert immer, sie ist aber nur in geeigneten Fallenleicht zu handhaben. Die universelle Derivation ist in der Regel nicht abschließ-bar, was aber fur Anwendungen in der Nichtkommutativen Geometrie irrele-vant ist. Wir erhalten immerhin ein hinreichendes, kohomologisches Kriterium

254 Michael Puschnigg

fur die Abschließbarkeit der universellen unbeschrankten Derivation. Diesesergibt sich aus dem Kotangentialkomplex [2, 2.9], muß aber mit großer Sorg-falt abgeleitet werden, da im Fall unbeschrankter Derivationen homologischeAlgebra mit “fast exakten” Folgen betrieben werden muß.

Durch Iteration wird die universelle Glattung C∞(S,A) von A bezuglich Sals Komplettierung des Graphen aller Potenzen der universellen Derivation dASkonstruiert. Der Kern des kanonischen Homomorphismus ι : C∞(S,A) → Aist topologisch nilpotent und sein Bild ist eine dichte, unter holomorphemFunktionalkalkul abgeschlossene Unteralgebra von A. Insbesondere sind dievon ι induzierten Abbildungen auf der K-Theorie Isomorphismen. All diesfolgt aus bekannten, in [1] ausfuhrlich behandelten Prinzipien.

Hauptgegenstand dieses Aufsatzes ist ein konkretes Beispiel: die universel-le unbeschrankte Derivation mit Definitionsbereich C[F2] auf der reduziertenC∗-Algebra C∗r (F2) der freien nichtabelschen Gruppe mit zwei Erzeugern. DerGruppenring C[F2] ist quasifrei im Sinn von Cuntz und Quillen [2, 3.3], wasdie universelle Derivation explizit beschreibbar macht. Andererseits ist dieAlgebra C∗r (F2) weit davon entfernt, nuklear zu sein, was dafur sorgt, daßder Abschluß C1(C[F2], C

∗r (F2)) des Gruppenrings unter der Graphennorm der

universellen Derivation eine sehr kleine glatte Unteralgebra von C∗r (F2) ist.Wir zeigen in unserem Hauptsatz, daß sich jede Spur auf C[F2] mit Tragerin nur endlich vielen Konjugationsklassen von F2 zu einer beschrankten Spurauf C1(C[F2], C

∗r (F2)) fortsetzen laßt. Das ist bei keiner der bisher bekannten

glatten Unteralgebren von C∗r (F2) der Fall und hat interessante Konsequenzen,die an anderer Stelle naher besprochen werden sollen.

Es bleibt noch hinzuzufugen, daß das eben genannte Ergebnis auch fur belie-bige wort-hyperbolische Gruppen Gultigkeit besitzt. Da die in diesem Aufsatzbenutzte Beweisidee aber nicht auf den allgemeineren Fall ubertragbar ist,bleibt die Behandlung desselben einer weiteren Arbeit vorbehalten.

Im Herbst 2008 feierte Joachim Cuntz seinen 60. Geburtstag. Seine Ide-en und Arbeiten zur Nichtkommutativen Geometrie haben meine eigene Ent-wicklung als Mathematiker stark beeinflußt. Ich widme ihm diese Arbeit inFreundschaft und Dankbarkeit.

2. Die universelle unbeschrankte Derivation

Bevor wir uns dem eigentlichen Gegenstand dieser Note zuwenden, mochtenwir an einige Begriffsbildungen erinnern. Sei A eine Frechetalgebra. Ein Fre-chetraum und A-Linksmodul M heißt Frechet-A-Linksmodul, falls die Modul-multiplikation A×M →M beschrankt ist. Die Begriffe eines Frechet-Rechts-bzw. Frechet-Bimoduls werden analog definiert. Wir nehmen im folgendenalle C- bzw. A-linearen Abbildungen zwischen Frechetraumen, Moduln undAlgebren als stetig (beschrankt) an. Fur Frechetraume V,W ist die MengeL(V,W ) der stetigen (beschrankten) linearen Abbildungen von V nach W innaturlicher Weise selbst ein Frechetraum. Ist M (bzw. N) ein Frechet-Rechts-(bzw. Frechet-Linksmodul) so heißt eine Abbildung ϕ : M × N → V in einen


Die universelle unbeschrankte Derivation 255

Frechetraum V A-bilinear, falls sie stetig (beschrankt), sowie C-bilinear istund daruber hinaus ϕ(ma, n) = ϕ(m, an) fur alle m ∈ M , n ∈ N , a ∈ A gilt.Es existiert ein (bis auf Isomorphie eindeutiger) Frechetraum M⊗AN , der innaturlicher Weise der Bedingung

(2.1) MorA−bilin(M ×N,V ) ∼= L(M⊗AN,V ),

fur jeden Frechetraum V genugt. Der Raum M⊗AN heißt das projektive Ten-sorprodukt von M und N uber A. Das projektive Tensorprodukt uber C wirdmit ⊗ bezeichnet.

Nun konnen wir uns Derivationen zuwenden. Wir erinnern daran, daß einebeschrankte Derivation auf einer unitalen Frechetalgebra A mit Werten in demFrechet-A-Bimodul M eine beschrankte lineare Abbildung D : A→M ist, dieder Leibniz-Regel

(2.2) D(aa′) = D(a)a′ + aD(a′), ∀a, a′ ∈ Agenugt. Die Menge Der(A,M) der beschrankten Derivationen auf A mit Wer-ten in M ist ein abgeschlossener linearer Unterraum von L(A,M) und daherselbst ein Frechetraum.

Der A-Bimodul Ω1A der Differentiale wird durch die naturliche Erweiterung

0 −−−−→ Ω1A −−−−→ A⊗A m−−−−→ A −−−−→ 0

von Frechet-A-Bimoduln definiert. Sie ist kontrahierbar als Erweiterung vonFrechet-A-Linksmoduln: eine kontrahierende Homotopie wird durch s0 : A →A⊗A, a 7→ a⊗ 1 und s1 : A⊗A→ Ω1A, a⊗ a′ 7→ a⊗ a′− aa′⊗ 1 gegeben. Wirzitieren den wohlbekannten

Satz 2.1. Sei A eine Frechetalgebra.

i) Die Abbildung

(2.4) dA : A→ Ω1A, a 7→ a⊗ 1− 1⊗ aist eine beschrankte Derivation auf A.

ii) Sie ist universell in dem Sinn, daß fur jeden Frechet-A-Bimodul M diekanonische Abbildung

(2.5) HomA−Bimod(Ω1A,M)∼=−−−−→ Der(A,M), ϕ 7−→ ϕ dA

ein Isomorphismus ist.

Im folgenden fixieren wir einen beschrankten Homomorphismus S → A vonFrechetalgebren mit dichtem Bild. Wir fassen Frechet-A-Bimoduln vermogedieses Homomorphismus auch als Frechet-S-Bimoduln auf.

Definition 2.2. Eine unbeschrankte Derivation aufAmit Definitionsbereich Sist eine beschrankte Derivation auf S mit Werten in einem Frechet-A-Bimodul.

Die Menge DerS(A,M) der unbeschrankten Derivationen aufAmit Definiti-onsbereich S und Werten in dem Frechet-A-Bimodul M ist ein abgeschlossenerlinearer Unterraum von L(S,M) und daher selbst ein Frechetraum.

Satz 2.3. Sei A eine Frechetalgebra.



i) Die Abbildung

(2.6) dAS : S → A⊗SΩ1S⊗SA, s 7→ 1⊗ ds⊗ 1

ist eine unbeschrankte Derivation auf A mit Definitionsbereich S.ii) Sie ist universell in dem Sinn, daß fur jeden Frechet-A-Bimodul M die

kanonische Abbildung

(2.7)HomA−Bimod(A⊗SΩ1S⊗SA,M)

∼=−−−−→ DerS(A,M),

ϕ 7−→ ϕ dASein Isomorphismus ist.

Der Beweis ergibt sich unmittelbar aus der Kette naturlicher Isomorphismen

DerS(A,M) ∼= HomS−Bimod(Ω1S,M) ∼= HomA−Bimod(A⊗SΩ1S⊗SA,M).

Man erkennt, daß DerA(A,M) = Der(A,M) und daß die naturliche Ver-gißabbildung Der(A,M) → DerS(A,M) einen kanonischen beschrankten A-Bimodulhomomorphismus

(2.8) j : A⊗SΩ1S⊗SA→ Ω1A

induziert.Die Nutzlichkeit der obigen Begriffsbildung hangt ganz davon ab, inwieweit

man die Struktur des Bimoduls A⊗SΩ1S⊗SA bestimmen kann.Wir wenden uns jetzt der Frage zu, unter welchen Bedingungen die univer-

selle unbeschrankte Derivation dAS abschließbar ist.

3. Der Kotangentialkomplex und die Abschließbarkeit deruniversellen Derivation

Zunachst mussen die kategoriellen Eigenschaften des projektiven Tensor-produkts genauer untersucht werden. Wir reformulieren hierzu einige bekann-te Begriffsbildungen aus der homologischen Algebra im Kontext topologischerVektorraume.

Wir nennen einen Kettenkomplex

C• : . . . −−−−→ Ci+1∂i+1−−−−→ Ci

∂i−−−−→ Ci−1 −−−−→ . . . , i ∈ Z

von Frechetraumen fast exakt, falls

(3.1) Ker(∂i) = Im(∂i+1)

fur alle i ∈ Z gilt. Die reduzierte Homologie eines Komplexes C∗ ist der gradu-ierte Vektorraum mit homogenen Komponenten

(3.2) Hi(C•) = Ker(∂i)/Im(∂i+1), i ∈ Z.

Jede Komponente ist in naturlicher Weise ein Frechetraum.



Lemma 3.1. Sei

0 −−−−→ M ′ −−−−→ M −−−−→ M ′′ −−−−→ 0

eine linear kontrahierbare Erweiterung von Frechet-A-Rechtsmoduln. Dann istfur jeden Frechet-A-Linksmodul N die Sequenz

(3.3) M ′⊗AN −−−−→ M⊗AN −−−−→ M ′′⊗AN −−−−→ 0

fast exakt.

Beweis: Betrachte das kommutative Diagramm

0 −−−−→ M ′⊗N −−−−→ M⊗N −−−−→ M ′′⊗N −−−−→ 0

π′

yyπ

yπ′′

M ′⊗AN −−−−→ M⊗AN −−−−→ M ′′⊗AN −−−−→ 0.

Die vertikalen Abbildungen sind surjektiv und die obere Zeile spaltet nach Vor-aussetzung. Das Lemma ergibt sich dann aus einer einfachen Diagrammjagd.

Ein Frechet-A-Linksmodul heißt frei, falls er zu einem Modul der FormA⊗V , V ein Frechetraum, isomorph ist. Er heißt projektiv, falls er zu ei-nem direkten Summanden eines freien Moduls isomorph ist. Eine projektiveAuflosung eines Frechet-A-Linksmoduls M ist ein linear kontrahierbarer Kom-plex

0 ←−−−− M ←−−−− P0 ←−−−− P1 ←−−−− . . . ,

bei dem P0, P1, . . . projektive Frechet-A-Moduln sind. Die entsprechenden Be-griffe fur Frechet-Rechts- oder Bimoduln werden analog eingefuhrt. Man kannin der ublichen Weise zeigen, daß jeder Frechet-A-Modul eine projektive Auf-losung besitzt und daß je zwei projektive Auflosungen eines gegebenen Modulskettenhomotopie-aquivalent sind. Dies ergibt sich daraus, daß jeder Homo-morphismus von Frechet-A-Moduln zu einer Kettenabbildung von gegebenenprojektiven Auflosungen fortgesetzt werden kann und diese Fortsetzung bis aufKettenhomotopie eindeutig bestimmt ist.

Definition 3.2. Sei (M,N) ein Paar von Frechet-A-(Rechts-Links)-Moduln.Sei (P•, Q•) → (M,N) eine projektive Auflosung von (M,N). Die Frechet-raume

(3.4) TorA

∗ (M,N) = H∗(P•⊗AQ•)werden als die Tor-Raume des Paares (M,N) uber A bezeichnet. Sie hangennicht von der Wahl der projektiven Auflosung ab und definieren kovarianteBifunktoren auf geeigneten Kategorien von Frechetmoduln.

Wir geben die funktoriellen Eigenschaften nicht im Detail an, da sie imfolgenden nicht von Bedeutung sind.

Proposition 3.3. Sei

0 −−−−→ M ′ −−−−→ M −−−−→ M ′′ −−−−→ 0



eine linear kontrahierbare Erweiterung von Frechet-A-Rechtsmoduln (Frechet-(A′, A)-Bimoduln). Sei M projektiv und sei N ein Frechet-A-Linksmodul (Fre-chet-(A,A′′)-Bimodul). Dann existiert eine naturliche, fast exakte Sequenz

(3.5) 0→ TorA

1 (M ′′, N)→M ′⊗AN →M⊗AN →M ′′⊗AN → 0

von Frechetraumen (Frechet-(A′, A′′)-Bimoduln). Fur n > 1 existieren kano-nische Isomorphismen

(3.6) TorA

n (M ′′, N)∼=−→ Tor

A

n−1(M′, N)

von Frechetraumen (Frechet-(A′, A′′)-Bimoduln).

Beweis: Ist dieses Ergebnis auch vollkommen “plausibel”, so ist im Umgangmit fast exakten Sequenzen doch Vorsicht geboten, da die meisten bekanntenelementaren Resultate aus der homologischen Algebra fur sie keine Gultigkeitmehr besitzen.

Seien P• → M ′, Q• → M ′′ projektive Auflosungen von M ′ und M ′′. Dannexistiert bekanntermaßen ein projektive Auflosung R• → M von M mit fol-genden Eigenschaften:i) Es existiert ein kommutatives Diagramm von Auflosungen mit linear kon-trahierbaren Zeilen

0 −−−−→ P•i•−−−−→ R•

p•−−−−→ Q• −−−−→ 0y

yy

0 −−−−→ M ′i−−−−→ M

p−−−−→ M ′′ −−−−→ 0.

ii) In jedem Grad spaltet die obere Zeile des Diagramms als Erweiterung vonFrechet-A-Moduln. In der Tat haben wir ein kommutatives Diagramm

0 −−−−→ Pnin−−−−→ Rn

pn−−−−→ Qn −−−−→ 0∥∥∥

y∥∥∥

0 −−−−→ Pni′−−−−→ Pn ⊕Qn p′−−−−→ Qn −−−−→ 0

von Erweiterungen von Frechet-A-moduln fur jedes n ∈ N.



Wenden wir darauf den Funktor −⊗AN an, so erhalten wir ein kommutati-ves Diagramm y

yy

0 −−−−→ P1⊗AN −−−−→ R1⊗AN −−−−→ Q1⊗AN −−−−→ 0y

yy

0 −−−−→ P0⊗AN −−−−→ R0⊗AN −−−−→ Q0⊗AN −−−−→ 0y

yy

M ′⊗AN i′−−−−→ M⊗AN p′−−−−→ M ′′⊗AN −−−−→ 0y

yy

0 0 0mit folgenden Eigenschaften:i) Alle Zeilen mit Ausnahme der untersten spalten als Erweiterungen vonFrechet-Raumen. Die unterste Zeile ist fast exakt.ii) Da M ein projektiver Frechet-A-Modul ist, ist die mittlere Spalte linearkontrahierbar.iii) Die linke und rechte Spalte sind exakt in niedrigstem Grad und fast exaktin zweitniedrigstem Grad.

Die Behauptung ergibt sich daraus durch eine langliche aber offensichtlicheDiagrammjagd.

Das Hauptergebnis dieses Abschnitts ist der

Satz 3.4. (Kotangentialkomplex [2]) Sei S → A ein Homomorphismus vonFrechetalgebren mit dichtem Bild. Dann existiert eine kanonische, fast exakteSequenz

(3.7) 0 −−−−→ TorS

1 (A,A) −−−−→ A⊗SΩ1S⊗SA −−−−→ Ω1A −−−−→ 0

von Frechetraumen.

Beweis: Betrachten wir die kanonische Erweiterung

0 −−−−→ Ω1S −−−−→ S⊗S m−−−−→ S −−−−→ 0

von Frechet-S-Bimoduln. Sie ist kontrahierbar als Erweiterung von Frechet-S-Linksmoduln und geht daher unter dem Funktor A⊗S− in die linear kontra-hierbare Erweiterung

0 −−−−→ A⊗SΩ1S −−−−→ A⊗S m−−−−→ A −−−−→ 0

von Frechet-A-Linksmoduln uber. Beachte, daß A⊗S frei und daher projek-tiv als Frechet-S-Rechtsmodul ist. Wenden wir den Funktor −⊗SA auf diegegebene Erweiterung an, so erhalten wir nach 3.3 die fast exakte Folge

0→ TorS1 (A,A)→ A⊗SΩ1S⊗SA→ A⊗A m→ A→ 0.



Daraus ergibt sich unmittelbar das gesuchte Ergebnis.

Eine Konsequenz dieses Resultats ist das

Korollar 3.5. Sei S → A ein Homomorphismus von Frechetalgebren mit dich-

tem Bild. Gilt TorS

1 (A,A) = 0, so ist die universelle unbeschrankte Derivationvon A mit Definitionsbereich S abschließbar.

Beweis: Sei (αn) eine Folge in S, die der Bedingung limn→∞

(αn, dAS (αn)) =

(0, β) ∈ A × A⊗SΩ1S⊗SA genugt. Sei j : A⊗SΩ1S⊗SA −→ Ω1A die ka-nonische Abbildung. Es gilt dann j(β) = j( lim

n→∞dAS (αn)) = lim

n→∞dA(αn) =

dA( limn→∞

αn) = dA(0) = 0, da dA stetig auf A ist. Unter den gegebenen Vor-

aussetzungen ist j injektiv und es folgt β = 0. Daraus ergibt sich unmittelbardie Behauptung.

4. Die universelle Glattung bezuglich einer dichtenUnteralgebra

Sei S → A ein Homomorphismus von Frechetalgebren mit dichtem Bildund sei dAS : S → A⊗SΩ1S⊗SA die universelle unbeschrankte Derivation vonA mit Definitionsbereich S. Wir bezeichnen mit C1(S,A) den Abschluß desGraphen von dAS in der Frechetalgebra A ⋉ A⊗SΩ1S⊗SA mit Multiplikation(a, ω) · (a′, ω′) = (aa′, aω′ + ωa′). Da dAS eine Derivation ist, wird C1(S,A)zu einer Frechetunteralgebra derselben. Die Projektion ι1 : C1(S,A) → A istein Homomorphismus von Frechetalgebren mit dichtem Bild und sein KernI = Ker(ι1) ist ein Ideal vom Quadrat Null: I2 = 0. Dieser Kern verschwindetgenau dann, wenn die universelle Derivation dAS abschließbar ist. Schließlichist die Abbildung S → C1(S,A), s 7→ (s, 1⊗ ds⊗ 1) ein Homomorphismus vonFrechetalgebren mit dichtem Bild. Dies erlaubt es die Konstruktion zu iterierenund man erhalt ein projektives System(4.1)

−−−−→ Cn(S,A)ιn−−−−→ Cn−1(S,A) −−−−→ . . . −−−−→ C1(S,A)

ι1−−−−→ A

von S-Frechetalgebren wobei

(4.2) Cn(S,A) = C1(S, Cn−1(S,A))

ist und die Strukturabbildungen Homomorphismen von S-Frechet-Algebrensind. Des weiteren besitzt der kanonischen Homomorphismus S → Cn(S,A) eindichtes Bild. Durch Ubergang zum projektiven Limes erhalten wir schließlichdie S-Frechetalgebra

(4.3) C∞(S,A) = lim←n

Cn(S,A)

Sie wird als die universelle Glattung von A bezuglich S bezeichnet. Es gilt die



Proposition 4.1. (Siehe [1]) Sei S → A ein Homomorphismus von Frechet-algebren mit dichtem Bild. Sei C∞(S,A) die universelle Glattung von A bezug-lich S und sei

(4.4) ι : C∞(S,A)→ A

der kanonische Homomorphismus.Dann gilt

i) Ist A zulassig im Sinn von [4, 1-3], so auch Cn(S,A), n ≥ 1, undC∞(S,A).

ii) Der Kern des kanonischen Homomorphismus ι ist ein topologisch nil-potentes Ideal in C∞(S,A).

iii) Das Bild des kanonischen Homomorphismus ι ist eine dichte, unterholomorphem Funktionalkalkul abgeschlossene Unteralgebra von A.

iv) Der kanonische Homomorphismus ι induziert Isomorphismen in topo-logischer K-Theorie und, falls A die Approximationseigenschaft vonGrothendieck besitzt, auch in (bivarianter) lokaler zyklischer Kohomo-logie.

Beweis: Behauptung i) ist im wesentlichen mit [4, 7.4] identisch. Sei I = Ker ι.Da der Kern von ιn : Cn(S,A) → Cn−1(S,A) ein Ideal vom Quadrat Null ist,muß das Bild von I2n in Cn(S,A) verschwinden. Daraus ergibt sich unmittel-bar die Aussage ii). Insbesondere stimmt das Spektrum eines Elements vonC∞(S,A) mit dem Spektrum seines Bildes in A unter dem kanonischen Homo-morphismus ι uberein. Dies impliziert iii) und die erste Aussage in iv) (siehe[1]). Die zweite Aussage in iv) ergibt sich schließlich aus [4, 7.1] oder [5], 5.15.

5. Ein Beispiel

Sei F2 die freie Gruppe mit den zwei kanonischen Erzeugern s, t. Die Grup-penalgebra C[F2] ist dann quasifrei im Sinn von Cuntz und Quillen [2, 5.3] undΩ1C[F2] ist frei als C[F2]-Bimodul [2, 3.3]. In der Tat ist

(5.1)

2⊕1

C[F2]⊗ C[F2]∼=−−−−→ Ω1C[F2]

a0 ⊗ a1 ⊕ b0 ⊗ b1 7−→ a0dusa1 + b0dutb

1

ein Isomorphismus von C[F2]-Bimoduln. Die analoge Behauptung gilt auch furden Bimodul der Differentiale uber der zulassigen Frechetalgebra ℓ1RD(F2) mitdefinierenden Seminormen ||∑ agug||k =

∑(1 + l(g))k|ag|, k ∈ N. (Hier be-

zeichnet l die kanonische Wortmetrik auf F2.) Wir betrachten nun die universel-le unbeschrankte Derivation auf der reduzierten Gruppen-C∗-Algebra C∗r (F2)mit Definitionsbereich C[F2]. Diese scheint a priori nicht in unseren Rahmenzu passen, da C[F2] keine Frechetalgebra ist. Der Einwand hat aber keinen Be-stand, da sich jede Derivation auf C[F2] mit Werten in einem C∗r (F2)-Bimodulzu einer beschrankten Derivation auf der Frechetalgebra ℓ1RD(F2) fortsetzenlaßt. Insbesondere ist also C1(C[F2], C

∗r (F2)) = C1(ℓ1RD(F2), C

∗r (F2)).



Lemma 5.1. Die universelle unbeschrankte Derivation d(C[F2], C∗r (F2)) auf

C∗r (F2) mit Definitionsbereich C[F2] ist abschließbar. Insbesondere istC1(C[F2], C

∗r (F2)) eine Unteralgebra von C∗r (F2) und d(C[F2], C

∗r (F2)) laßt sich

zu einer beschrankten Derivation

(5.2) δ : C1(C[F2], C∗r (F2)) → C∗r (F2)⊗C[F2] Ω

1C[F2]⊗C[F2]C∗r (F2) = Ω′(F2)

fortsetzen.

Beweis: Sei (αn) eine Folge in C[F2], fur die

limn→∞

(αn, d(C[F2], C∗r (F2))(αn)) = (0, β) ∈ C∗r (F2)× Ω′(F2)

gilt. Nach [3, 1.8] existiert ein Netz (ϕα) von Funktionen mit endlichen Tragerauf F2 , so daß die zugehorigen (punktweisen) Multiplikationsoperatoren Mϕα

auf C∗r (F2) die Bedingungen

||Mϕα || ≤ 1, ∀α, und lim→α

Mϕαx = x, ∀x ∈ C∗r (F2),

erfullen. Sei M ′ϕα der beschrankte Operator auf Ω′(F2), der unter der Iden-

tifizierung (5.1) dem Operator Mϕα⊗Mϕα auf2⊕1C∗r (F2)⊗C∗r (F2) entspricht.

Man sieht dann leicht, daß

lim→n

M ′ϕα(d(C[F2], C∗r (F2))(αn)) = M ′ϕα(β) = 0, ∀α,

und daher β = lim→α

M ′ϕα(β) = 0 ist, woraus die Behauptung folgt.

Ziel dieser Arbeit ist der folgende Satz, der die Existenz einer sehr “kleinen”holomorph abgeschlossenen Unteralgebra in C∗r (F2) behauptet. Ein entsprech-endes Resultat gilt auch fur beliebige wort-hyperbolische Gruppen. Da es abermit anderen Methoden gewonnen wird, bleibt seine Darstellung einer weiterenArbeit vorbehalten.

Satz 5.2. Sei C[F2] der komplexe Gruppenring der freien Gruppe mit zwei Er-zeugern. Sei C∗r (F2) die reduzierte Gruppen-C∗-Algebra von F2 und seiC1(C[F2], C

∗r (F2)) die zu der universellen unbeschrankten Derivation

mit Definitionsbereich C[F2] assoziierte Unteralgebra. Fur jede Konjugations-klasse 〈x〉 ⊂ F2 bezeichne τ〈x〉 die durch die charakteristische Funktion von 〈x〉definierte Spur auf C[F2]. Dann gilt

i) Die zulassige Frechetalgebra C1(C[F2], C∗r (F2)) ist dicht und abgeschlos-

sen unter holomorphem Funktionalkalkul in C∗r (F2).ii) Fur jede Konjugationsklasse 〈x〉 in F2 laßt sich das Funktional τ〈x〉 zu

einer beschrankten Spur auf C1(C[F2], C∗r (F2)) fortsetzen.

Beweis: Die universelle unbeschrankte Derivation auf C∗r (F2) mit Definitions-bereich C[F2] nimmt Werte in dem freien Bimodul Ω′(F2) (5.2) an. Eine Basis



desselben wird nach (5.1) durch

(5.3)

2⊕1C∗r (F2)⊗C∗r (F2)

∼=−−−−→ Ω′(F2)

a0 ⊗ a1 ⊕ b0 ⊗ b1 7−→ a0dusa1 + b0dutb

1

gegeben. Im folgenden werden wir ohne weitere Erwahnung von dieser Identi-fizierung Gebrauch machen.

Sei nun 〈x〉 6= 1 eine Konjugationsklasse in F2 (unsere Behauptung ist imFall 〈x〉 = 1 evident). Mit Mult(〈x〉) bezeichnen wir die großte naturliche Zahln, fur die ein (und damit jedes) Element von 〈x〉 eine n-te Potenz in F2 ist.Wir fixieren ein Element g minimaler Wortlange in 〈x〉. Nach eventueller Umbe-nennung der Erzeuger s, t, s−1, t−1 durfen wir annehmen, daß das minimaleWort fur g mit dem Buchstaben s endet. Wir betrachten nun die beschrankteAbbildung

(5.4)µg : Ω′(F2) −−−−→ C

a0dusa1 + b0dutb

1 7−→ 〈ξg, πreg(a1a0us)ξe〉.

Die Komposition

(5.5) µg δ : C1(C[F2], C∗r (F2))

δ−→ Ω′(F2)µg−→ C

definiert ein beschranktes lineares Funktional auf C1(C[F2], C∗r (F2)). Wir be-

haupten, daß seine Einschrankung auf den Gruppenring C[F2] der Identitat

(5.6) µg δ = Mult(〈x〉) · τ〈x〉genugt, woraus sich unmittelbar der Satz ergibt. Dazu betrachten wir ein Ele-ment der Form uh ∈ C[F2], h ∈ F2. Man sieht leicht, daß µx(uh) = 0, fallsh nicht zu x konjugiert ist. Sei also h ∈ 〈x〉. Wir wahlen ein Element mi-nimaler Wortlange v ∈ F2, so daß das reduzierte Wort fur g′ = v−1hv ei-ne zyklische Permutation des reduzierten Wortes fur g ist. Dann ergibt sich(µg δ)(uh) = µg(duvg′v−1) = µg(duvug′v−1)+µg(uvdug′uv−1)+µg(uvg′duv−1).Schreibt man duvug′v−1 = a0dusa1 + b0dutb1 so ist a1a0us =

∑λh′uh′ , wo-

bei die Wortlange der Elemente h′ ∈ F2 großer als die Wortlange von gist. Daher gilt µg(duvug′v−1) = 0 und aus denselben Grunden findet manµg(uvg′duv−1) = 0. Also ist (µg δ)(uh) = µg(uvdug′uv−1) = µg(dug′). Derletztere Ausdruck stimmt aber mit der Anzahl zyklischer Permutationen desreduzierten Wortes fur g′ uberein, die das reduzierte Wort fur g liefern. Diesist genau die Multiplizitat der Konjugationsklasse 〈x〉 und Gleichung (5.6) istbewiesen.

References

[1] B. Blackadar and J. Cuntz, Differential Banach algebra norms and smooth subalgebrasof C∗-algebras, J. Operator Theory 26 (1991), no. 2, 255–282. MR1225517 (94f:46094)

[2] J. Cuntz and D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8

(1995), no. 2, 251–289. MR1303029 (96c:19002)



[3] U. Haagerup, An example of a nonnuclear C∗-algebra, which has the metric approxima-tion property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR0520930 (80j:46094)

[4] M. Puschnigg, Asymptotic cyclic cohomology, Lecture Notes in Math., 1642, Springer,Berlin, 1996. MR1482804 (99e:46098)

[5] M. Puschnigg, Diffeotopy functors of ind-algebras and local cyclic cohomology, Doc.Math. 8 (2003), 143–245 (electronic). MR2029166 (2004k:46128)

Received February 2, 2009; accepted February 17, 2009

Michael PuschniggInstitut de Mathematiques de Luminy, UMR 6206 du CNRSUniversite de la Mediterranee, 163, Avenue de Luminy, 13288 Marseille, Cedex 9, FranceE-mail: [email protected]




Dirac operators for coadjoint orbits of compact

Lie groups

Marc A. Rieffel


In celebration of the 60th birthday of Joachim Cuntz

Abstract. The coadjoint orbits of compact Lie groups carry many Kahler structures, whichinclude a Riemannian metric and a complex structure. We provide a fairly explicit formulafor the Levi-Civita connection of the Riemannian metric, and we use the complex structure togive a fairly explicit construction of a canonical Dirac operator for the Riemannian metric,in a way that avoids use of the spinc groups. Substantial parts of our results apply tocompact almost-Hermitian homogeneous spaces, and to other connections besides the Levi-Civita connection. For these other connections we give a criterion that is both necessaryand sufficient for their Dirac operator to be formally self-adjoint.

We hope to use the detailed results given here to clarify statements in the literatureof high-eneregy physics concerning “Dirac operators” for matrix algebras that converge tocoadjoint orbits. To facilitate this we employ here only global methods—we never use localcoordinate charts, and we use the cross-section modules of vector bundles.

Introduction

In the literature of theoretical high-energy physics one finds statementsalong the lines of “matrix algebras converge to the sphere” and “here arethe Dirac operators on the matrix algebras that correspond to the Dirac op-erator on the sphere”. But one also finds that at least three inequivalenttypes of Dirac operator are being used in this context. See, for example,[2, 1, 4, 6, 11, 20, 21, 41, 42] and the references they contain, as well as [30]which contains some useful comparisons. In [34, 35, 38] I provided definitionsand theorems that give a precise meaning to the convergence of matrix algebrasto spheres. These results were developed in the general context of coadjointorbits of compact Lie groups, which is the appropriate context for this topic, asis clear from the physics literature. I seek now to give a precise meaning to the

The research reported here was supported in part by National Science Foundation Grants

DMS-0500501 and DMS-0753228.

266 Marc A. Rieffel

statement about Dirac operators. For this purpose it is important to have adetailed understanding of Dirac operators on coadjoint orbits, in a form that iscongenial to the noncommutative geometry that is used in treating the matrixalgebras. This means, for example, that one should work with the modules ofcontinuous sections of vector bundles, rather than the points of the bundlesthemselves, and one should not use local coordinate charts. (Standard moduleframes are very useful to us in this connection.) The purpose of this paper is togive such a congenial detailed understanding of Dirac operators on coadjointorbits.

Let G be a connected compact semisimple Lie group, with Lie algebra g. Letg′ denote the vector-space dual of g, and let µ ∈ g′ with µ 6= 0. The coadjointorbit of µ can be identified with G/K where K is the stability subgroup of µ.Then µ determines a G-invariant Kahler structure on G/K, which includes aRiemannian metric and a complex structure [8]. This complex structure deter-mines a canonical spinc structure on G/K. A principal objective of this paperis to give a reasonably explicit construction of the Dirac operator for this spinc

structure. Toward this objective we obtain in Section 3 a reasonably specificformula for the Levi-Civita connection for the Riemannian metric determinedby µ. (The only place I have seen this Levi-Civita connection discussed in theliterature is in Section 7 of [5], where the context is not sufficiently congenialto noncommutative geometry for my purposes.) Our construction of the Diracoperator, along the lines given in [33, 19, 39], never involves the spinc groups,with their attendant complications. We will also consider Dirac operators forspinc structures obtained by twisting the canonical one.

We remark that coadjoint orbits are always spinc manifolds, but many arenot spin manifolds. See [5, 32, 14, 27] for interesting specific examples. But Ihave not found a description in the literature of exactly which coadjoint orbitsare spin. (Though see Remark 3.6 of [18].) We will not discuss here the chargeconjugation that can be constructed for the Dirac operator coming from a spinstructure, but underlying the spin structure on a coadjoint orbit that is spinwill be one of the twisted spinc structures that we consider, for the reasonsindicated by Definition 9.8 of [19].

But there are otherG-invariant metrics of interest onG/K, the most obviousone coming from using the Killing form of g. This metric will come from theKahler structure on a coadjoint orbit only in the special case that G/K is asymmetric space. More generally, as is explained well on page 21 of [15], if theLevi-Civita connection for a Riemannian manifold commutes with a complexstructure, then the Riemannian metric is part of a Kahler structure on themanifold. But as explained in [8], if G/K has a Kahler structure then G/Kmust correspond to a coadjoint orbit. The consequence of this is that if wewant to treat Riemannian metrics such as that from the Killing form, and if wewant to use a complex (or almost-complex) structure to construct the Diracoperator, then we must use connections that are not torsion-free. But then wemust be concerned with whether the corresponding Dirac operator is formallyself-adjoint, as is usually desired.


Dirac operators for coadjoint orbits of compact Lie groups 267

To deal with this more general situation, we develop a substantial part ofour results for the more general case in which G/K is almost-Hermitian. Thereare many more coset spaces G/K that admit a G-invariant almost-Hermitianstructure, beyond those that arise from coadjoint orbits. In Theorem 6.1 wegive a convenient criterion, in terms of the torsion, that is both necessaryand sufficient, for the Dirac operator constructed using a connection compati-ble with a G-invariant almost-Hermitian structure, to be formally self-adjoint.Our criterion is very similar to the one given in the main theorem of [23],which treats the case of homogeneous spaces that are spin. (See also [17].)The criterion in [23] is restated as Proposition 3.1 of [3], which again treatshomogeneous spaces that are spin, and focuses on “naturally reductive” Rie-mannian metrics. As we will indicate after Theorem 3.3, the metric from theKahler structure of a coadjoint orbit is “naturally reductive” exactly in thespecial case when the coadjoint orbit is a symmetric space. Also, our globaltechniques are different from the techniques of these two papers.

Among the corollaries of our criterion we prove that for the canonical con-nection on an almost-HermitianG/K its Dirac operator is always formally self-adjoint. In particular, this applies to coadjoint orbits when they are equippedwith the Riemannian metric coming from the Killing form. (In this case thecanonical connection often has nonzero torsion.)

It would be very interesting to know how the results in the present paperrelate to those in [26]. In [26] only one “metric” on a quantum flag manifoldappears to be used, and my guess is that it corresponds to the Killing-form met-ric, and that the self-adjointness of the Dirac operator relates to our Corollary6.6. But I have not studied this matter carefully. It would also be interestingto study the extent to which the results of the present paper can be extendedto the setting of [14], or used in the setting immediately after Equation 6.31of [28].

The present paper builds extensively on the paper [37], in which I gavea treatment of equivariant vector bundles, connections, and the Hodge-Diracoperator, for general G/K with G compact, in a form congenial to the frame-work of noncommutative geometry. (The most recent arXiv version of [37] hasimportant corrections and improvements compared to the published version.)

In Section 1 of the present paper we describe at the level of the Lie algebrathe Kahler structure for a coadjoint orbit. In Section 2 we obtain a generalformula for the Levi-Civita connection for a G-invariant Riemannian metric ona coset space G/K for G compact. In Section 3 we use results from Section 1together with the general formula of Section 2 to obtain a rather specific for-mula for the Levi-Civita connection for the Riemannian metric of the Kahlerstructure on a coadjoint orbit. At no point do we need to use the full struc-ture theory of semisimple Lie algebras—we only need the nondegeneracy ofthe Killing form. In Section 4 we develop, at the level of the Lie algebra, theClifford algebra and its spinor representation corresponding to the complexstructure of an almost-Hermitian coset space; and then in Section 5 we usethis to define the field of Clifford algebras, the spinor bundle, and the Dirac


268 Marc A. Rieffel

operator for an almost-Hermitian coset space G/K. We also obtain there someof the basic properties of the Dirac operator. Finally, in Section 6 we obtainthe criterion mentioned above for when the Dirac operator will be formallyself-adjoint, and we apply this criterion to the case of the Riemannian metricfrom the Kahler structure of a coadjoint orbit, and also to the case of theRiemannian metric from the Killing form.

A part of the research for this paper was carried out during a six-week visit Imade to Scuola Internazionale Superiore di Studi Avanzati (SISSA) in Trieste,where Dirac vibrations are strong. I am very appreciative of the stimulatingatmosphere there, and of the warm hospitality of Gianni Landi and LudwikDabrowski during my visit.

I am very grateful to the referee for detailed comments on the first versionof this paper, which in particular led to some important improvements.

1. The canonical Kahler structure

Let G be a connected compact Lie group. Let g be its Lie algebra, and letAd be the adjoint action of G on g. Let g′ be the vector-space dual of g, andlet Ad′ be the coadjoint action of G on g′, that is, the dual of the action Ad.The coadjoint orbits are the orbits in g′ for the action Ad′. Let µ⋄ ∈ g′, withµ⋄ 6= 0. We will obtain in this section quite explicit formulas for the restrictionto the tangent space at µ⋄ of the canonical Kahler structure on the coadjointorbit through µ⋄. We will usually mark with a ⋄ the various pieces of structurethat depend canonically on the choice of µ⋄. In Sections 2 and 3 we will seehow to construct the Kahler structure on the whole coadjoint orbit through µ⋄.This Kahler structure includes a Riemannian metric and a complex structure.In Section 5 we will construct the Dirac operator for this Riemannian metricon the canonical spinc structure determined by the complex structure.

Since the center of G leaves all the points of g′ fixed, we do not lose generalityby assuming that G is semisimple. We assume this from now on. But we willsee that the only aspect of semisimplicity that we will need is the definitenessof the Killing form. We do not need the structure theory of semisimple Liealgebras.

For much of the material in this section I have been guided by the contentsof [8]. In [8] many possibilities are explored. In contrast, we will here tryto take the shortest path to what we need, and we will emphasize the extentto which the structures are canonical. We will not examine what happenswhen we choose different µ⋄’s that have the same stability group. But [8] hasconsiderable discussion of this aspect.

Let K denote the Ad′-stability subgroup of µ⋄, so that x 7→ Ad′x(µ⋄) givesa G-equivariant diffeomorphism from G/K onto the Ad′-orbit of µ⋄. We willusually work with G/K rather than the orbit itself.

We let Kil denote the negative of the Killing form of g. Then Kil is positive-definite because G is compact. The action Ad of G on g is by orthogonaloperators with respect to Kil, and the action ad of g on g is by skew-adjoint



operators with respect to Kil. Because Kil is definite, there is a Z⋄ ∈ g suchthat

(1.1) µ⋄(X) = Kil(X,Z⋄) for all X ∈ g.

It is easily seen that the Ad-stability subgroup of Z⋄ is again K.Let T⋄ be the closure in G of the one-parameter group r 7→ exp(rZ⋄), so

that T⋄ is a torus subgroup of G. Then it is easily seen that K consists exactlyof all the elements of G that commute with all the elements of T⋄. Note thatT⋄ is contained in the center of K (but need not coincide with the center).Since each element of K will lie in a torus subgroup of G that contains T⋄, itfollows that K is the union of the tori that it contains, and so K is connected(Corollary 4.22 of [24]). Thus for most purposes we can just work with the Liealgebra, k, of K when convenient. In particular, k = X ∈ g : [X,Z⋄] = 0,and k contains the Lie algebra, t⋄, of T⋄.

Let m = k⊥ with respect to Kil. Since Ad preserves Kil, we see that m iscarried into itself by the restriction of Ad to K. Thus [k,m] ⊆ m. It is well-known, and explained in [37], that m can be conveniently identified with thetangent space to G/K at the coset K (which corresponds to the point µ⋄ of thecoadjoint orbit). We will review this in the next section. Here we concentrateon the structures on m that will give the Kahler structure on G/K.

The Kahler structure includes a symplectic form ω⋄. This is the Kirillov-Kostant-Souriau form, defined initially on g by

(1.2) ω⋄(X,Y ) = µ⋄([X,Y ]) = Kil([X,Y ], Z⋄) = Kil(Y, [Z⋄, X ]).

Because Z⋄ is in the center of k, we see that if X ∈ k then ω⋄(X,Y ) = 0 for allY ∈ g. Conversely, if X ∈ g and if ω⋄(X,Y ) = 0 for all Y ∈ g, then, becauseKil is nondegenerate, we have [X,Z⋄] = 0, so that X ∈ k. Thus ω⋄ “lives” onm and is nondegenerate there. Because Ad preserves Kil and K stabilizes Z⋄,it is easily seen that the restriction of Ad to K preserves ω⋄, that is,

ω⋄(Ads(X),Ads(Y )) = ω⋄(X,Y )

for all X,Y ∈ m and s ∈ K.We now follow the proof of Proposition 12.3 of [10] in order to construct

a complex structure on m. (I am grateful to Xiang Tang for bringing thisproposition to my attention. My original, somewhat longer, approach at thispoint was to begin working in the complexification of g and m, as done in[8].)See also the proof of Theorem 1.36 of [9] and the middle of the second proofof Proposition 2.48i of [29]. Because Kil is nondegenerate, there is a uniquelinear operator, Γ⋄, on m such that

(1.3) ω⋄(X,Y ) = Kil(Γ⋄X,Y )

for all X,Y ∈ m. From Equation 1.2 we see that Γ⋄ is adZ⋄restricted to m, and

so Γ⋄ is skew-symmetric, that is, Γ∗⋄ = −Γ⋄. Because ω⋄ is nondegenerate, Γ⋄ isinvertible. Because Z⋄ is in the center of k, the Ad-action of K commutes withΓ⋄. Let Γ⋄ = |Γ⋄|J⋄ be the polar decomposition of Γ⋄. Since Γ⋄ is invertible,so are |Γ⋄| and J⋄, and thus J⋄ is an orthogonal transformation with respect


270 Marc A. Rieffel

to Kil. Because Γ⋄ is skew-symmetric, so is J⋄, so that J−1⋄ = J∗⋄ = −J⋄,

and J⋄ commutes with |Γ⋄|. In particular, J2⋄ = −I, where I denotes the

identity operator on m. This means exactly that J⋄ is a complex structure onm, preserved by the Ad-action of K.

The final piece of structure is a corresponding inner product, g⋄, on m,defined by

g⋄(X,Y ) = ω⋄(X, J⋄Y ) = Kil(Γ⋄X, J⋄Y ) = Kil(|Γ⋄|X,Y ).

Clearly g⋄ is positive-definite, and is preserved by the Ad-action of K. It is g⋄that will give the Riemannian metric whose Dirac operator we will construct.The complex structure J⋄ will enable us to avoid the use of spinc groups whenconstructing the Dirac operator.

But first we need to obtain a reasonably explicit expression for the Levi-Civita connection for the Riemannian metric corresponding to g⋄. For thispurpose we need to examine the Ad-action of T⋄ on m. By means of J⋄ wemake m into a C-vector-space, by defining iX to be just J⋄X for X ∈ m. Whenwe view m as a C-vector-space in this way we will denote it by mJ⋄

. Since theAd-action of K (and thus of T⋄) on m commutes with J⋄, this action respects

the C-vector-space structure. We define a C-sesquilinear inner product, KilC⋄ ,on m by

KilC⋄ (X,Y ) = Kil(X,Y ) + iKil(J⋄X,Y ).

It is linear in the second variable. (We follow the conventions in Definition 5.6of [19].) The Ad-action of K on mJ⋄

is unitary for this inner product. The Ad-action of T⋄ on mJ⋄

then decomposes into a direct sum of one-dimensional com-plex representations of T⋄, whose corresponding representations of t⋄ are givenby real-linear functions on t⋄ whose values are pure-imaginary (the “weights”of the ad-action). We let ∆⋄ be the set of real-valued linear functionals α ont⋄ such that iα is a weight of the ad-action. It will be convenient for us to set,for each real-linear real-valued functional α on t⋄,

mα = X ∈ mJ⋄: adZ(X) = iα(Z)X = α(Z)J⋄X for all Z ∈ t⋄.

Thus mα = 0 exactly when α /∈ ∆⋄. For any X ∈ mα and Y ∈ mJ⋄we see

from Equation 1.2 that

g⋄(X,Y ) = ω⋄(X, J⋄Y ) = Kil([Z⋄, X ], J⋄Y )

= Kil(α(Z⋄)J⋄X, J⋄Y ) = α(Z⋄)Kil(X,Y ).

Thus for α ∈ ∆⋄ and X ∈ mα with X 6= 0 we have

0 < g⋄(X,X) = α(Z⋄)Kil(X,X),

and so α(Z⋄) > 0. Thus in terms of the above notation we see that we obtainthe following attractive description of |Γ⋄|:Proposition 1.4. For each α ∈ ∆⋄ the restriction of |Γ⋄| to mα is α(Z⋄)Imα ,where Imα is the identity operator on mα. In particular, α(Z⋄) > 0, and onmα we have g⋄ = α(Z⋄)Kil. If Pα denotes the orthogonal projection of m ontomα, then |Γ⋄| =

∑α∈∆⋄

α(Z⋄)Pα.



Note that this proposition shows how strongly dependent g⋄ is on the choiceof µ⋄. In contrast, different µ⋄’s that give Z⋄’s that generate the same groupT⋄ may have the same subspaces mα.

2. Levi-Civita connections for invariant Riemannian metricson G/K

In this section we assume as before that G is a connected compact semisim-ple Lie group, but we only assume that K is a closed subgroup of G, notnecessarily connected. We will assume that we have an inner-product, g0, onm that is invariant under the Ad-action of K. We do not assume that g0 isthe restriction to m of an Ad-invariant inner product on g, as was assumedin [37]. We will see shortly that much as in [37], g0 determines a G-invariantRiemannian metric on G/K. We seek a formula for the Levi-Civita connectionfor this metric. On m there is a positive (for Kil) invertible operator, S, suchthat g0(X,Y ) = Kil(SX, Y ) for all X,Y ∈ m. (So S for a coadjoint orbit isthe |Γ0| of the previous section.) Note that S commutes with the Ad-actionof K. Our formula will be expressed in terms of S. In Section 3 we will usethis formula to obtain a more precise formula for the Levi-Civita connectionfor a coadjoint orbit. Toward the end of this section we will also discuss thedivergence theorem for vector fields on G/K. We need this for our discussionof the formal self-adjointness of Dirac operators in Section 6.

As in [37], we work with the module of tangent vector fields. For brevitywe will at times refer to such “induced” modules as “bundles”. We recall thesetting here. We let A = C∞R (G/K), which we often view as a subalgebra ofC∞R (G). The tangent bundle of G/K is

T (G/K) = V ∈ C∞(G,m) : V (xs) = Ad−1s (V (x)) for x ∈ G, s ∈ K.

It is an A-module for the pointwise product, and G acts on it by translation.We denote this translation action by λ. Each V ∈ T (G/K) determines aderivation, δV , of A by

(δV f)(x) = Dt0(f(x exp(tV (x))),

where Dt0 means “derivative in t at t = 0”. On T (G/K) we have the canonical

connection, ∇c, defined by

(2.1) (∇cV (W ))(x) = Dt0(W (x exp(tV (x)))

for V, W ∈ T (G/K). It is not in general torsion-free. Associated to it is the“natural torsion-free” [25] connection, ∇ct, that is given (e.g. in Theorem 6.1of [37]) by

∇ct = ∇c + Lct,

where LctV for any V ∈ T (G/K) is the A-module endomorphism of T (G/K)defined by

(2.2) (LctVW )(x) = (1/2)P [V (x),W (x)],


272 Marc A. Rieffel

where P is the projection of g onto m along k. Then ∇ct is the Levi-Civitaconnection for the case in which g0 is the restriction of Kil to m. Both ∇c and∇ct are G-invariant in the sense suitable for connections [37].

Our given inner product g0 determines a Riemannian metric on G/K, alsodenoted by g0, defined by

(g0(V,W ))(x) = g0(V (x),W (x))

for all V,W ∈ T (G/K) and x ∈ G. Thus g0(V,W ) ∈ A. When there isno ambiguity about the choice of g0 we will often write 〈V,W 〉A instead ofg0(V,W ) ∈ A. This Riemannian metric is G-invariant (and every G-invariantRiemannian metric arises in this way). We seek to adjust ∇ct to obtain theLevi-Civita connection, ∇0, for g0. A convenient method for doing this is givenby Theorem X.3.3 of [25] (or Equation 13.1 of [31], where there is a sign error).We seek ∇0 in the form ∇ct+LS, where LS is an A-linear map from T (G/K)into the A-endomorphisms of T (G/K). We require that LS be symmetric, thatis that LSWV = LSVW for all V,W ∈ T (G), since this ensures that ∇0 is torsionfree, because ∇ct is. As seen in [37], by translation invariance we can calculateat x = e, the identity element of G. Then according to Theorem X.3.3 of[25] we are to determine the symmetric bilinear form Φ on m that satisfies theEquation

(2.3) 2g0(Φ(X,Y ), Z) = g0(X,P [Z, Y ]) + g0(P [Z,X ], Y )

for all X,Y, Z ∈ m. For the reader’s convenience we recall the reasoning. Forx = e we have LctX(Y ) = (1/2)P [X,Y ] for X,Y ∈ m. Set LS on m to beLSX(Y ) = Φ(X,Y ). Then the above equation becomes

g0(LSXY, Z) = g0(X,L

ctZY ) + g0(L

ctZX,Y ).

When we add to this equation its cyclic permutation

g0(LSZX,Y ) = g0(Z,L

SYX) + g0(L

SY (Z), X)

and use the symmetry of g0 and Φ and the fact that LctZY = −LctY Z, we obtain

g0(LSXY, Z) + g0(Y, L

SXZ) = −g0(LctXY, Z)− g0(Y, LctXZ).

This says exactly that the operator LctX +LSX on m is skew-symmetric with re-spect to g0. This implies that when LS is extended to T (G/K) by G-invariance(in the sense that λx(L

SVW ) = LSλxV λxW as discussed in Section 5 of [37]) the

connection ∇ct+LS is compatible with the Riemannian metric g0 (as seen, forexample, from Corollary 5.2 of [37]). This connection is also torsion-free, andthus it is the Levi-Civita connection for g0.

When we rewrite Equation 2.3 in terms of Kil and S we obtain

2 Kil(SΦ(X,Y ), Z) = Kil(SX,P [Z, Y ]) + Kil(P [Z,X ], SY )

= Kil([Y, SX ], Z) + Kil(Z, [X,SY ]).

Since this must hold for all Z, we see that

LSXY = Φ0(X,Y ) = (1/2)S−1P ([X,SY ] + [Y, SX ]).



By G-invariance as above

(2.4) (LSVW )(x) = (1/2)S−1P ([V (x), SW (x)] + [W (x), SV (x)])

for V,W ∈ T (G/K) and x ∈ G. We thus obtain:

Theorem 2.5. The Levi-Civita connection for the Riemannian metric g0 is∇0 = ∇ct + LS where LS is defined by (2.4) and S relates g0 to Kil as above.

Let ∆ denote the set of eigenvalues of S, and for each α ∈ ∆ let mα denotethe corresponding eigensubspace. For α, β ∈ ∆ and X ∈ mα, Y ∈ mβ we seethat

(2.5) LSXY = (1/2)S−1P ([X, βY ] + [Y, αX ]) = (1/2)(β − α)S−1P [X,Y ],

and thus the complication in getting a more precise formula lies in expressingS−1P [X,Y ] in terms of the eigensubspaces of S. In Section 3 we will see howto obtain such a more precise formula for the case of coadjoint orbits.

But first we derive here a form of the divergence theorem for our vectorfields, because we will need it in Section 6, and Equation (2.4) is important forits proof. We recall from [37] that by a standard module frame for T (G/K)with respect to the Riemannian metric g0 we mean a finite collection Wj ofelements of T (G/K) that have the reproducing property

V =∑

Wj〈Wj , V 〉Afor all V ∈ T (G/K). (We view T (G/K) as a right A-module, following theconventions in [19].)

Definition 2.7. Let ∇0 be the Levi-Civita connection for the Riemannianmetric g0 on G/K. We define the divergence, div(V ), of an element V ∈T (G/K), with respect to g0, to be

(2.8) div(V ) =∑

j

g0(∇0WjV,Wj),

where Wj is a standard module frame for T (G/K).

It is not difficult to check that this definition coincides with the usual def-inition of the divergence in terms of differential forms, but we do not needthis fact here. We should make sure that our definition is independent of thechoice of the frame Wj. To prove our divergence theorem we actually needa slightly more general form of frames, so we give the independence argumentin terms of these. The argument is essentially well-known.

Proposition 2.9. Let A be a commutative ring and let E be an A-module thatis equipped with an A-valued symmetric bilinear form 〈·, ·〉A. Assume that thereexist biframes for E with respect to this bilinear form, that is, there are finitesets (Wj , Wj) of pairs of elements of E such that V =

∑Wj〈Wj , V 〉A for

every V ∈ E. Then for any A-bilinear form β on E, not necessarily symmetric,with values in some A-module, the sum

∑j β(Wj , Wj) is independent of the

choice of biframe.


274 Marc A. Rieffel

Proof. Let (Uk, Uk) be another biframe. Then∑

j

β(Wj , Wj) =∑

j

∑

k,l

β(Uk〈Uk,Wj〉A, Ul〈Ul, Wj〉A)

=∑

k,l

β(Uk, Ul)∑

j

〈Uk,Wj〈Wj , Ul〉A〉A

=∑

k

β(Uk,∑

l

Ul〈Ul, Uk〉A) =∑

k

β(Uk, Uk).

This proof can be made more conceptual by noting that 〈·, ·〉A establishes anisomorphism of E ⊗A E with EndA(E).

Our greater generality is needed because we want to use frames that involvethe fundamental vector fields X, for X ∈ g, that correspond to the action ofG by translation on G/K. As shown in Section 4 of [37], they are given by

X(x) = −P Ad−1x (X).

It is also shown in Section 4 of [37] that if Xj is an orthonormal basis for g

for Kil, then Xj is a standard module frame for the Riemannian metric onG/K coming from restricting Kil to m. Thus for any V ∈ T (G/K) we have

V =∑

j

Xj Kil(Xj , V ) =∑

Xjg0(S−1Xj , V ).

From this we see that the collection (Xj, S−1Xj) is a biframe for T (G/K)

when T (G/K) is equipped with g0. On G/K we use the G-invariant measurecoming from a choice of Haar measure on G.

Theorem 2.10. Let g0 be a G-invariant Riemannian metric on G/K and letdiv(V ) be defined as above for g0. Then for any V ∈ T (G/K) we have

∫

G/K

div(V ) = 0.

Proof. We have ∇0 = ∇c + Lct + LS . We split∫G/K

div(V ) into the corre-

sponding three terms, and show that each is 0. The first term is∫

G/K

∑

j

g0(∇cWjV, Wj).

It is independent of the choice of frame Wj by Proposition 2.9, and by thatproposition we can, in fact, use the biframe defined just above. Now ∇c iscompatible with g0, and so

g0(∇cXjV, S−1Xj) = δXj (g0(V, S

−1Xj))−∑

g0(V,∇cXj (S−1Xj)).

But as discussed in the proof of Theorem 8.4 of [37], for any X ∈ g andany f ∈ A we have

∫G/K δX(f) = 0, because δX(f) is the uniform limit of

the quotients (λexp(−tX)f − f)/t as t → 0, and the integral of each of these



quotients is 0 by the G-invariance of the measure on G/K. Thus we see thatwe would like to show that∫

G/K

g0(V,∑

j

∇cXj

(S−1Xj)) = 0.

For that it suffices to show that∑

j

∇cXj

(S−1Xj) = 0.

But ∇c only involves derivatives, and since S−1 is constant, it is clear fromEquation 2.1 that S−1 commutes with ∇c. It thus suffices to show that∑∇c

XjXj = 0. This was shown at the end of the proof of Theorem 8.4 in

[37]. We recall the reasoning here. Early in Section 6 of [37] it is shown thatfor each X,Y ∈ g

(∇cXY )(x) = −P ([P Ad−1

x (X),Ad−1x (Y )])

for all x ∈ G. By Proposition 2.9 for each fixed x ∈ G we can choose the basisXj such that Ad−1

x (Xj) is the union of a Kil-orthonormal basis for m and

one for k. For such a basis (∇cXjXj)(x) = 0 for each j.

Now let L0 = Lct + LS . It remains to show that∫

G/K

∑

j

g0(L0WjV,Wj) = 0

for each V ∈ T (G/K). But∇0 = ∇c+L0, and ∇0 is assumed to be compatiblewith g0. Consequently each L0

U is skew-adjoint for g0. Thus

g0(L0WjV,Wj) = −g0(V, L0

WjWj),

and so we see that it suffices to show that∑

j L0WjWj = 0. To show this we

treat Lct and LS separately. Now from Equation 2.2 we see that for each j

(LctWjWj)(x) = (1/2)P [Wj(x),Wj(x)] = 0.

Thus∑

j LctWjWj = 0

Finally, from Equation 2.4 we see that

(∑

j

LSWjWj)(x) = (1/2)S−1P

∑

j

[Wj(x), SWj(x)] + [Wj(x), SWj(x)]

= S−1P∑

j

[Wj(x), SWj(x)].

But Wj(x) is a frame for m and g0, for each x ∈ G, and by Proposition2.9 the above expression is independent of the chosen frame. Notice that Sis positive for g0 as well as for Kil. Consequently as frame we can choose ag0-orthonormal basis for m consisting of eigenvectors of S. It is then clear that(∑

j LSWjWj)(x) = 0.


276 Marc A. Rieffel

3. The Levi-Civita connection for coadjoint orbits

We now return to the setting of coadjoint orbits as in Section 1, with S =|Γ⋄|. We will obtain here the more precise formula for |Γ⋄|−1P [X,Y ] thatTheorem 2.5 indicates we need in order to obtain a precise formula for theLevi-Civita connection for g⋄. Motivation for some of the expressions that weconsider can be found by working in the complexification of g along the linesused in [8]. For any α, β ∈ ∆⋄ set |α − β| = α − β if (α − β)(Z⋄) ≥ 0, andotherwise set |α−β| = β−α, so that always |α−β|(Z⋄) ≥ 0. Of course |α−β|may not be in ∆⋄. Recall from Proposition 1.4 that if γ ∈ ∆⋄ then γ(Z⋄) > 0.We do not have [m,m] ⊆ m, but nevertheless:

Lemma 3.1. Let α, β ∈ ∆⋄, and let X ∈ mα and Y ∈ mβ. Then

[X,Y ]− [J⋄X, J⋄Y ] ∈ mα+β (so = 0 if α+ β /∈ ∆⋄),

while

[X,Y ] + [J⋄X, J⋄Y ] ∈

k if α = β

m|α−β| if α 6= β (so = 0 if |α− β| /∈ ∆⋄).

Thus, on adding, we find that

[X,Y ] ∈

mα+β ⊕m|α−β| if α 6= β

m2α ⊕ k if α = β .

Furthermore,

J⋄([X,Y ]− [J⋄X, J⋄Y ]) = [J⋄X,Y ] + [X, J⋄Y ],

while if α 6= β then

J⋄([X,Y ] + [J⋄X, J⋄Y ]) = sign(α(Z⋄)− β(Z⋄))([J⋄X,Y ]− [X, J⋄Y ]).

If α = β then J⋄P ([X,Y ] + [J⋄X, J⋄Y ]) = 0.

Proof. Note that [X,Y ] need not be in m. Let Z ∈ t. Within the calculationsbelow we will, for brevity, often write just α for α(Z) and similarly for β. Thenfrom the Jacobi identity we have

adZ([X,Y ]− [J⋄X, J⋄Y ])

= [αJ⋄X,Y ] + [X, βJ⋄Y ] + [αX, J⋄Y ] + [J⋄X, βY ]

= (α+ β)([J⋄X,Y ] + [X, J⋄Y ]).

On substituting J⋄X for X in the equation above we obtain

adZ([J⋄X,Y ] + [X, J⋄Y ]) = −(α+ β)([X,Y ]− [J⋄X, J⋄Y ]),

and on combining these two equations we obtain

(adZ)2([X,Y ]− [J⋄X, J⋄Y ]) = −(α+ β)2([X,Y ]− [J⋄X, J⋄Y ]).

Recall that adZ carries m into itself and sends k to 0, so the range of adZis in m. Now let Z = Z⋄, so that α > 0 and β > 0. Then we see from the abovecalculations that ([X,Y ] − [J⋄X, J⋄Y ]) ∈ m. Recall also that adZ⋄

= |Γ⋄|J⋄,



so that (adZ⋄)2 = −|Γ⋄|2. Then from the above calculations it becomes clear

that [X,Y ]− [J⋄X, J⋄Y ] ∈ mα+β . Of course it may be that α+β /∈ ∆⋄ so thatmα+β = 0. From the above calculations we see furthermore that

J⋄([X,Y ]− [J⋄X, J⋄Y ]) = [J⋄X,Y ] + [X, J⋄Y ].

In the same way, for any Z ∈ t we have

adZ([X,Y ] + [J⋄X, J⋄Y ]) = (α− β)([J⋄X,Y ]− [X, J⋄Y ])

andadZ([J⋄X,Y ]− [X, J⋄Y ]) = (β − α)([X,Y ] + [J⋄X, J⋄Y ]),

so that

(adZ)2([X,Y ] + [J⋄X, J⋄Y ]) = −(α− β)2([X,Y ] + [J⋄X, J⋄Y ]).

If α = β then it is clear from these calculations that ([X,Y ] + [J⋄X, J⋄Y ]) ∈k. If α 6= β, then on letting Z = Z⋄ and arguing as above, we see that([X,Y ] + [J⋄X, J⋄Y ]) ∈ m|α−β| for the definition of |α− β| given above. Thestatement about J⋄([X,Y ] + [J⋄X, J⋄Y ]) now follows much as before.

Recall the definition of LS from Equations 2.4 and 2.5. We now use theabove lemma to obtain a more precise formula for LS for the present case inwhich S = Γ⋄. We denote this LS for Γ⋄ by L⋄.

Proposition 3.2. Let α, β ∈ ∆⋄, and let X ∈ mα and Y ∈ mβ. Then

4L⋄XY = (β(Z⋄)− α(Z⋄))(α(Z⋄) + β(Z⋄))−1([X,Y ]− [J⋄X, J⋄Y ])

+ sign(β(Z⋄)− α(Z⋄))([X,Y ] + [J⋄X, J⋄Y ]),

as long as we make the convention that sign(0) = 0.

Proof. From Lemma 3.1 we see that

|Γ⋄|−1P [X,Y ]

= (1/2)|Γ⋄|−1P (([X,Y ]− [J⋄X, J⋄Y ]) + ([X,Y ] + [J⋄X, J⋄Y ]))

= (1/2)((α(Z⋄) + β(Z⋄))−1([X,Y ]− [J⋄X, J⋄Y ])

+ |α(Z⋄)− β(Z⋄)|−1P ([X,Y ] + [J⋄X, J⋄Y ]),

where the last term must be taken to be 0 if α = β. On substituting this intoEquation 2.5 and simplifying, we obtain the desired expression for L⋄X(Y ).

Recall now that the Levi-Civita connection for g⋄ is ∇⋄ = ∇ct+L⋄ = ∇c +Lct+L⋄, where on m we have LctXY = (1/2)P [X,Y ]. If we set L⋄t = Lct+L⋄,then from Proposition 3.2 we see that on m we have

4L⋄tXY

= (1 + (β(Z⋄)− α(Z⋄))(α(Z⋄) + β(Z⋄))−1)([X,Y ]− [J⋄X, J⋄Y ])

+ (1 + sign(β(Z⋄)− α(Z⋄)))P ([X,Y ] + [J⋄X, J⋄Y ]),

= 2β(Z⋄)(α(Z⋄) + β(Z⋄))−1([X,Y ]− [J⋄X, J⋄Y ])

+ (1 + sign(β(Z⋄)− α(Z⋄)))P ([X,Y ] + [J⋄X, J⋄Y ]).


278 Marc A. Rieffel

When we extend this to T (G/K) byG-invariance, and let T α(G/K) denote thesubspace of T (G/K) consisting of elements whose range is in mα, we obtain:

Theorem 3.3. The Levi-Civita connection ∇⋄ for the Riemannian metric g⋄is given for V ∈ T α(G/K) and W ∈ T β(G/K), for α, β ∈ ∆⋄, by

(∇⋄VW )(x) = (∇cVW )(x)

+ (1/4)(2β(Z⋄)(α(Z⋄) + β(Z⋄))

−1([V (x),W (x)] − [J⋄V (x), J⋄W (x)])

+ (1 + sign(β(Z⋄)− α(Z⋄)))P ([V (x),W (x)] + [J⋄V (x), J⋄W (x)]))

for all x ∈ G.

The above formula should be compared to Formula 7.15 in [5]. We remarkthat from Theorem 3.3 and Lemma 3.1 it is easily seen that g⋄ is “naturallyreductive” [25, 3], so has Levi-Civita connection equal to ∇ct [3], exactly whenG/K is a symmetric space, that is, when [m,m] ⊆ k

In our Kahler situation we expect that J⋄ will commute with ∇⋄. This isessential for the construction that we will give shortly for the Dirac operatorfor g⋄. We now check this fact directly.

Proposition 3.4. With notation as above, J⋄ commutes with ∇⋄.Proof. It is easily seen that J⋄ commutes with ∇c, so we only need to showthat it commutes with L⋄t. Note that in general J⋄ does not commute withLct, so we need to work with the combination Lct+L⋄ = L⋄t. By G-invarianceit suffices to deal just with elements of m. Let α, β ∈ ∆⋄, and let X ∈ mα andY ∈ mβ . For brevity we again often write just α for α(Z⋄) and similarly for βwithin our calculations. Then when we apply J⋄ to L⋄t and apply the resultsof Lemma 3.1, we obtain

4J⋄L⋄tXY = 2β(α+ β)−1([J⋄X,Y ] + [X, J⋄Y ])

+ (1 + sign(β − α))sign(α − β)P ([J⋄X,Y ]− [X, J⋄Y ]),

while4L⋄tX(J⋄Y ) = 2β(α+ β)−1([J⋄X,Y ] + [X, J⋄Y ])

+ (1 + sign(β − α))P ([X, J⋄Y ]− [J⋄X,Y ]).

Notice that

(1 + sign(β − α))sign(α− β)P ([J⋄X,Y ]− [X, J⋄Y ])

= (1 + sign(β − α))sign(β − α)P ([X, J⋄Y ]− [J⋄X,Y ])

= (sign(β − α) + 1)P ([X, J⋄Y ]− [J⋄X,Y ]).

Thus J⋄L⋄tXY = L⋄tX(J⋄Y ) as desired.

4. The spinor representation

In view of the results of the previous sections, it is appropriate to considerin general an even-dimensional real vector space m with a given inner productg0, a compact Lie group K that is not required to be semisimple or connected,a representation π (instead of Ad |K) of K on m preserving g0, and a complex



structure J on m respecting both g0 and π. For use in constructing a Diracoperator we seek a representation of the complex Clifford algebra over m for g0that respects the action of K. (Many coadjoint orbits are not spin manifolds[5, 32, 14, 27], only spinc.) Much of the material in this section is taken fromChapter 5 of [19]. The exposition in [19] is especially suitable for our needs,and it includes much detail on a number of aspects. But as before, here wewill try to take the shortest path to what we need. An important point is thatwe will find that because of the complex structure we do not need to involvethe spinc groups, with their attendant complexities.

As in [27, 19], we will denote the complex Clifford algebra over m for g0 byCℓ(m). It is the complexification of the real Clifford algebra for m and g0. Wefollow the convention that the defining relation is

XY + Y X = −2g0(X,Y )1.

We include the minus sign for consistency with [27, 37]. Thus in applying theresults of the first pages of Chapter 5 of [19] we must let the g there to be −g0.After Exercise 5.6 of [19] it is assumed that g is positive, so small changes areneeded when we use the later results in [19] but with our different convention.The consequence of including the minus sign is that in the representationswhich we will construct the elements of m will act as skew-adjoint operators,just as they do for orthogonal or unitary representations of G if m arises asin the previous section, rather than as self-adjoint operators as happens whenthe minus sign is omitted.

Because m is of even dimension, the algebra Cℓ(m) is isomorphic to a fullmatrix algebra [19]. We equip Cℓ(m) with the involution ∗ (conjugate linear,with (ab)∗ = b∗a∗) that takes X to −X for X ∈ m (again so that the elementsof m are skew-adjoint).

Let O(m, g0) denote the group of operators on m orthogonal for g0. By theuniversal property of Clifford algebras each element R of O(m, g0) determinesan automorphism of Cℓ(m) (a “Bogoliubov” automorphism) given on a productX1 · · ·Xp of elements of m in Cℓ(m) by

(4.1) R(X1X2 · · ·Xp) = R(X1)R(X2) · · ·R(Xp).

In this way we obtain a homomorphism from O(m, g0) into the automorphismgroup of Cℓ(m). Since π gives a homomorphism of K into O(m, g0) we obtaina homomorphism, still denoted by π, of K into the automorphism group ofCℓ(m), which extends the action of K on m. The Lie algebra so(m, g0) ofO(m, g0) will then act as a Lie algebra of derivations of Cℓ(m), given for L ∈so(m, g0) by

(4.2)L(X1X2 · · ·Xp) = L(X1)X2 · · ·Xp

+X1L(X2)X3 · · ·Xp + · · ·+X1 · · ·Xp−1L(Xp).

Corresponding to this we have an action of k as derivations of Cℓ(m), againdenoted by π.


280 Marc A. Rieffel

Because Cℓ(m) is isomorphic to a full matrix algebra, it has, up to equiva-lence, exactly one irreducible representation. We seek an explicit constructionof such a representation in a form which makes manifest that this representa-tion carries an action of K that is compatible with the action of K on Cℓ(m).As shown in [19] beginning with Definition 5.6, the complex structure J on m

leads to an explicit construction. (See also the discussion after Corollary 5.17of [27].) We will denote the resulting Hilbert space for this representation byS, for “spinors”.

To begin with, we use J to view m as a complex vector space by settingiX = JX , as we did earlier. We then define a positive-definite sesquilinearform, i.e., complex inner product, on m, by

〈X,Y 〉J = g0(X,Y ) + ig0(J(X), Y ).

Note that, as in [19], we take it linear in the second variable. When we viewm as a complex vector space with this inner product, we denote it by mJ .We note that because π commutes with J and preserves g0, it is a unitaryrepresentation of K on mJ (so that, in particular, actually π(K) ⊆ SO(m, g0)).As in Definition 5.7 of [19] we let F(mJ) denote the complex exterior algebra∧∗

mJ over mJ . It is referred to in [19] as the (unpolarized) Fock space. It willbe our space S of spinors, and we will write F(mJ ) or S as convenient. Thenwe equip S with the inner product determined by

(4.3) 〈X1 ∧ · · · ∧Xp, Y1 ∧ · · · ∧ Yq〉J = δpq det[〈Xk, Yl〉J ],

which is Equation 5.17a of [19]. Let U(mJ) denote the unitary group of mJ . Bythe universal property of exterior algebras the action of U(mJ) on mJ extendsto an action on F(mJ) by exterior-algebra automorphisms, defined in much thesame way as in Equation (4.1). By means of the homomorphism π from K intoU(mJ) we obtain an action of K as automorphism of F(mJ ), again denoted byπ. Then the Lie algebra u(mJ) of U(mJ) will act as a Lie algebra of exterior-algebra derivations of F(mJ), and by this means we obtain an action, π, of k

as derivations of F(mJ).We need a representation of Cℓ(m) on S. As done shortly after Exercise

5.12 of [19], we define annihilation and creation operators, aJ (X) and a†J (X),on F(mJ) for X ∈ m by

aJ (X)(X1 ∧ · · · ∧Xp) =

p∑

j=1

(−1)j−1〈X,Xj〉JX1 ∧ · · · ∧ Xj ∧ · · · ∧Xp

(where Xj means to omit that term), and

a†J (X)(X1 ∧ · · · ∧Xp) = X ∧X1 ∧ · · · ∧Xp

for X1, . . . , Xp ∈ mJ . Note that aJ(X) is conjugate linear in X . One thenchecks, much as done in the paragraph before Definition 5.1 of [19], that

aJ(X)a†J (Y ) = a†J(Y )aJ(X) = 〈X,Y 〉JIS ,



where IS is the identity operator on F(mJ), and

aJ(X)aJ (Y ) + aJ (Y )aJ (X) = 0 = a†J(X)a†J(Y ) + a†J(Y )a†J (X)

for X,Y ∈ m. We then set

κJ(X) = i(aJ(X) + a†J(X)).

(So the i here reflects our sign convention, different from that of [19].) Notethat κJ(X) is only real-linear in X . Using the anti-commutation relationsabove, we see that

κJ(X)κJ(Y ) + κJ(Y )κJ (X) = −〈X,Y 〉J − 〈Y,X〉J = −2g0(X,Y )

for X,Y ∈ m, where we omit IS on the right as is traditional. But this is therelation that defines Cℓ(m). Thus κJ extends by universality to give a homo-morphism, again denoted by κJ , from Cℓ(m) into the algebra, L(F(mJ )), oflinear operators on F(mJ). Let dimR(m) = 2n. Then dimC(mJ ) = n, so thatdimC(F(mJ)) = 2n. But dimC(Cℓ(m)) = 22n = (2n)2. Since Cℓ(m) is isomor-phic to a full matrix algebra, and since κJ is clearly not the 0 homomorphism,the homomorphism κJ must be bijective, and gives an irreducible representa-tion of Cℓ(m) on F(mJ). Thus we can take S = F(mJ) as our Hilbert spaceof spinors, with the action of Cℓ(m) on S given by κJ .

Recall that we have actions of O(m) on Cℓ(m) and of U(mJ) on F(mJ).Since U(mJ) ⊂ SO(m), we have an action of U(mJ) on Cℓ(m). Let us denoteby ρ the actions of U(mJ) on both Cℓ(m) and F(mJ ). A crucial fact for us is:

Proposition 4.4. The action κJ of Cℓ(m) on F(mJ) respects the actions ρ ofU(mJ) on Cℓ(m) and F(mJ) in the sense that

(4.5) ρR(κJ (c)ψ) = κJ (ρR(c))(ρR(ψ))

for all R ∈ U(mJ), c ∈ Cℓ(m) and ψ ∈ S.Proof. It suffices to show that

ρR(κJ (X)(X1 ∧ · · · ∧Xp)) = κJ(ρR(X))(ρR(X1 ∧ · · · ∧Xp))

for all R ∈ U(mJ) and all X,X1, . . . , Xp ∈ mJ . Now

ρR(a†J (X)(X1 ∧ · · · ∧Xp)) = ρR(X ∧X1 ∧ · · · ∧Xp)

= (R(X)) ∧ (R(X1)) ∧ · · · ∧ (R(Xp))

= a†J(R(X))(ρR(X1 ∧ · · · ∧Xp)).

A similar calculation, using the fact that ρ preserves 〈·, ·〉J , shows that

ρR(aJ (X)(X1 ∧ · · · ∧Xp)) = aJ(R(X))(ρR(X1 ∧ · · · ∧Xp)).

In view of how κJ is defined in terms of a†J and aJ , we see that (4.5) holds.

Since π carries K into U(mJ), we immediately obtain:

Corollary 4.6. The actions π of K on Cℓ(m) and F(mJ) are compatible withthe action κJ of Cℓ(m) on F(mJ ) in the sense given above.


282 Marc A. Rieffel

Let ρ denote also the actions of the Lie algebra u(mJ) on Cℓ(m) and F(mJ)by derivations. We quickly obtain the following corollary, which we will needlater for our discussion of connections:

Corollary 4.7. The action κJ of Cℓ(m) on F(mJ ) is compatible with theactions ρ of u(mJ) on Cℓ(m) and F(mJ ) in the sense of the Leibniz rule

(4.8) ρL(κ(c)ψ) = κ(ρL(c))ψ + κ(c)ρL(ψ)

for all L ∈ u(mJ), c ∈ Cℓ(m) and ψ ∈ S.Notice that we have never needed to use explicitly the spinc groups in our

discussion.Next, in order to see that everything fits well, let us show that κJ respects

the involutions, where by this we mean that

(4.9) κJ(c∗) = (κJ(c))∗

for all c ∈ Cℓ(m), where the ∗ on the left is the involution on Cℓ(m) definedearlier, while the ∗ on the right means the adjoint of the operator for the innerproduct on S. (Thus S is a “self-adjoint Clifford module” as in Definition 9.3of [19], but for our conventions.) It suffices to prove this for c = X for allX ∈ m, that is, it suffices to show that (κJ(X))∗ = −κJ(X). In view of how

κJ is defined in terms of aJ and a†J it suffices to show that

(aJ (X))∗ = a†J(X).

This is well-known, and can be seen as follows. We can assume that ‖X‖J = 1.Set e1 = X , and choose e2, . . . , en ∈ mJ such that e1, . . . , en is an orthonormalC-basis for mJ . For any I = j1 < j2 < · · · < jp ⊆ 1, . . . , n set eI =ej1 ∧ ej2 ∧ · · · ∧ ejp in S, and set e∅ = 1. A glance at (4.3) shows that eI isan orthonormal basis for S. Then from (4.3) one quickly sees that

〈aJ(e1)eI1 , eI2〉J = 1 if 1 ∈ I1 and I2 = I1 \ 1,and is 0 otherwise, while

〈eI1 , a†J(e1)eI2〉J = 1 if 1 /∈ I2 and I1 = I2 ∪ 1,and is 0 otherwise. This shows that a†J (e1) is the adjoint of aJ(e1).

Finally, let us consider the chirality element, following the discussion inDefinition 5.2 of [19] and the paragraphs following it. Choose an orientationfor m, and let X1, . . . , X2n be an oriented orthonormal R-basis for m and g0.Define the chirality element, γ, of Cℓ(m) (for the chosen orientation) by

γ = (i)nX1X2 · · ·X2n.

(The i is included because our sign convention differs from that of [19].) Then,much as discussed in [19], γ does not depend on the choice of the orientedorthonormal basis, and it satisfies γ2 = 1, γ∗ = γ and γXγ = −X for everyX ∈ m. In particular, conjugation by γ is the grading operator on Cℓ(m)that gives the even and odd parts. Since U(mJ) is connected, each elementof U(mJ) carries an oriented orthonormal basis into an other one, and thus



leaves γ invariant. Consequently, for any X ∈ u(mJ) its derivation action onCℓ(m) takes γ to 0. Since π(K) ⊆ U(mJ) we have πs(γ) = γ for all s ∈ K,and πX(γ) = 0 for all X ∈ k. Because κJ is a ∗-representation, we will have(κJ (γ))2 = 1 and κJ (γ) = (κJ(γ))∗. Since γ 6= 1, κJ(γ) 6= IS , and thusκJ(γ) will split S into two orthogonal subspaces, S±, the “half-spinor” spaces.Because πs(γ) = γ for all s, each of S+ and S− will be carried into itself bythe representation π of K on S. Because γXγ = −X for X ∈ m, we see thatκJ(X) will carry S+ into S− and S− into S+ for each X ∈ m. Of course eachof S+ and S− will be carried into itself by the subalgebra of even elements ofCℓ(m).

5. Dirac operators for almost-Hermitian G/K

In this section we assume as before that G is a compact semisimple Liegroup, but we only assume that K is a closed subgroup of G, not necessarilyconnected. We assume further only that G/K is (homogeneous) almost Her-mitian, by which we mean that we have an inner product, g0, on m that isinvariant for the Ad-action of K on m, and that we have a complex structureJ on m that is orthogonal for g0 and commutes with the Ad-action of K (som is even-dimensional). Then g0 and J are extended to T (G/K) pointwise.There are many examples of such coset spaces beyond the coadjoint orbits.See for example the many constructions in Sections 8 and 9 of [40]. But I havenot seen in the literature any complete classification of all of the possibilitiesfor almost-Hermitian compact coset spaces. In this section we show how toconstruct a “Dirac operator” for any connection on T (G/K) that is compatiblewith g0 and commutes with J . For coadjoint orbits we have seen that boththe canonical connection and (in Theorem 3.4) the Levi-Civita connection ∇⋄for g⋄ commute with J⋄.

Much as in Section 7 of [37] we can form the Clifford bundle over G/K forg0, except that here we use the complex Clifford algebra that was discussed inthe previous section instead of the real Clifford algebra used in [37]. The roleof π of the previous section is now taken by Ad restricted to K and acting onm, and so also on Cℓ(m) = Cℓ(m, g0). From now on we will denote this action

of K on Cℓ(m) by Ad. We set(5.1)

Cℓ(G/K) = c ∈ C∞(G,Cℓ(m)) : c(xs) = Ad−1

s (c(x)) for x ∈ G, s ∈ K.

It is clearly an algebra for pointwise operations. We let AC = C∞(G/K,C),and we see that not only is Cℓ(G/K) an algebra over AC, but that in fact AC

can be identified with the center of Cℓ(G/K) (since m is even-dimensional).Furthermore, Cℓ(G/K) contains the tangent bundle T (G/K) of G/K as a real(generating) subspace. On Cℓ(G/K) we have the action λ of G by translation,and this action defines the canonical connection, ∇c, on Cℓ(G/K), which actsby derivations (much as discussed in [37]). Clearly this ∇c extends the ∇c onT (G/K).


284 Marc A. Rieffel

Suppose now that ∇ is some other connection on T (G/K) that is G-invariant and compatible with g0 (such as our earlier ∇⋄ when G/K is acoadjoint orbit). As seen in Section 5 of [37], especially Corollary 5.2, ∇ isthen of the form ∇ = ∇c + L where L is a G-equivariant A-homomorphismfrom T (G/K) into EndskA (T (G/K)). Here EndskA (T (G/K)) denotes the A-endomorphisms of T (G/K) that are skew-adjoint with respect to g0. As seenin Proposition 3.1 of [37], each such endomorphism LV for V ∈ T (G/K) isgiven by a smooth function on G whose values are in so(m, g0), which we denoteagain by LV , and which satisfies the condition

LV (xs) = Ad−1s LV (x) Ads

for x ∈ G and s ∈ K. For any V,W ∈ T (G/K) we have (LVW )(x) =(LV (x)(W (x)) for x ∈ G. By Equation (4.2) each LV will extend to a deriva-tion of Cℓ(G/K), and in this way we obtain an A-linear (so R-linear) map fromT (G/K) into the Lie-algebra of derivations of Cℓ(G/K). (These derivationswill, in fact, be ∗-derivations for the involution determined by the involution onCℓ(m) defined in Section 4.) We can now define a G-invariant connection, ∇,on Cℓ(G/K) by ∇ = ∇c+L. It clearly extends the original∇ on T (G/K) (andδ on AC). Again ∇V will be a derivation of Cℓ(G/K) for each V ∈ T (G/K).Note that our construction of Cℓ(G/K) and its ∇ does not use J .

We use J in the way described in the previous section to define the complexHilbert space S = F(mJ) of spinors, with its compatible actions of Cℓ(m) and

K. We will again denote the action of K on S by Ad. We then define thecanonical bundle S(G/K) of spinor fields on G/K for J by

(5.2) S(G/K) = ψ ∈ C∞(G,S) : ψ(xs) = Ad−1

s (ψ(x)) for x ∈ G, s ∈ K.It is an AC-module in the evident way (projective by Proposition 2.2 of [37]).

As explained in Theorem 1.7i of [33] and Proposition 9.4 of [19] and laterpages, spinor bundles for Clifford bundles are not in general unique. The tensorproduct of a spinor bundle by a line bundle will be another spinor bundle, andall the spinor bundles are related in this way. Within our setting of equivariantbundles we need to tensor with G-equivariant line bundles. These correspondexactly to the characters, that is, one-dimensional representations, of K. (Wewill not discuss Dirac operators twisted by vector bundles of higher dimension.)From Theorem 5.1 of [37] it is easily seen that G-invariant connections on a linebundle differ from the canonical connection by a constant. For our purposes wecan ignore the constant. In fact, even the canonical connection need not appearexplicitly. We proceed as follows. Let χ be a character of K. (We remark thatwhen G/K is a coadjoint orbit, K always has nontrivial characters because t

is an ideal in k.) We set:

S(G/K,χ) = ψ ∈ C∞(G,S) : ψ(xs) = χ(s)Ad−1

s (ψ(x)) for x ∈ G, s ∈ K.On S(G/K,χ) we define an AC-valued inner product in the usual way by

〈ψ, ϕ〉AC(x) = 〈ψ(x), ϕ(x)〉S



for ψ, ϕ ∈ S(G/K,χ). Of greatest importance is the action κ of Cℓ(G/K) onS(G/K,χ) that is defined by

(5.3) (κ(c)ψ)(x) = κ(c(x))(ψ(x))

for x ∈ G. (We drop the subscript J on κ used in the previous section.) Thisaction carries S(G/K,χ) into itself because

(κ(c)ψ)(xs) = (κ(Ad−1

s (c(x))))(χ(s)Ad−1

s (ψ(x))) = χ(s)Ad−1

s ((κ(c)ψ)(x)),

where the last equality follows from Proposition 4.4.On S(G/K,χ) we have the action λ of G by translation, and it is easily

seen that κ is compatible with this action and the G-action on Cℓ(G/K). Theaction λ defines a canonical connection on S(G/K,χ) by adapting (2.1) in theevident way. We will denote this canonical connection again by ∇c. Of primeimportance, we have the Leibniz rule

(5.4) ∇cV (κ(c)ψ) = κ(∇cV c)ψ + κ(c)(∇cV ψ)

for any V ∈ T (G/K), c ∈ Cℓ(G/K) and ψ ∈ S(G/K,χ). Furthermore, muchas discussed in [37], the connection on S(G/K,χ) is compatible with the AC-valued inner product in the sense of the Leibniz rule

(5.5) δV (〈ψ, ϕ〉AC) = 〈∇cV ψ, ϕ〉AC

+ 〈ψ,∇cV ϕ〉AC

for any V ∈ T (G/K) and ψ, ϕ ∈ S(G/K,χ). The only property of J that isused for this is the evident fact that when we view J as acting on T (G/K)pointwise, it commutes with the translation action of G.

Suppose now that our original∇ on T (G/K) commutes with J , in the sensethat each∇V does. As before, set∇ = ∇c+L. Since also∇c commutes with J ,each LV will commute with J , that is, LV (x) ∈ u(mJ) for each x ∈ G. Then,as discussed in the previous section, each LV (x) will extend to a derivationof the exterior algebra S = F(mJ), and consequently LV determines an A-module endomorphism of S(G/K,χ), which we denote by LSV . In this waywe define a G-equivariant A-linear map LS from T (G/K) into the algebra ofAC-endomorphisms of S(G/K,χ). Furthermore, it is easily checked that eachLSV is skew-adjoint for the AC-valued inner product on S(G/K,χ). Of mostimportance, we see from Corollary 4.7 that LS is compatible with the actionof Cℓ(G/K) on S(G/K,χ) in the sense of the Leibniz rule

LSV (κ(c)ψ) = κ(LV c)ψ + κ(c)(LSV ψ)

for all V , c and ψ. We saw in (5.4) that ∇c satisfies a similar identity, and sowe have obtained:

Proposition 5.6. Let (G/K, g0, J) be almost Hermitian, and let χ be a char-acter of K. Let ∇ be a G-invariant connection on T (G/H) that is compatiblewith g0 and commutes with J . Let ∇ also denote its extension to a connec-tion on Cℓ(G/K) as constructed above, and let ∇S denote the correspondingconnection on S(G/K,χ) constructed above using J . Then the Leibniz rule

∇SV (κ(c)ψ) = κ(∇V c)ψ + κ(c)(∇SV ψ)


286 Marc A. Rieffel

holds for all V ∈ T (G/K), c ∈ Cℓ(G/K) and ψ ∈ S(G/K,χ). Furthermore,∇S is compatible with the AC-valued inner product on S(G/K,χ) from g0.

In terms of the connection ∇S on S(G/K,χ) from ∇ we can define the“Dirac” operator for ∇ (which when G/K is a coadjoint orbit will be thecanonical Dirac operator for µ⋄ when χ is trivial, S(G/K) is constructed us-ing J⋄, and ∇⋄ is the Levi-Civita connection for g⋄). Much as done in Sec-tion 8 of [37], for any ψ ∈ S(G/K,χ) we define dψ by dψ(V ) = ∇SV (ψ) forV ∈ T (G/K). Then we can view dψ as an element of T ∗(G/K)⊗RS(G/K,χ),where T ∗(G/K) denotes the A-module of smooth cross-sections of the cotan-gent bundle. By means of the Riemannian metric g0 (as A-valued inner prod-uct) we can identify T ∗(G/K) with T (G/K). When dψ is viewed by thisidentification as an element of T (G/K)⊗R S(G/K,χ) we will denote it, withsome abuse of notation, by grad0 ψ. Let us view the Clifford action κ ofCℓ(G/K) on S(G/K,χ) as a bilinear mapping from Cℓ(G/K) ⊗C S(G/K,χ)into S(G/K,χ). We can view T (G/K) as a real subspace of Cℓ(G/K) inthe evident way, and so we can view T (G/K) ⊗R S(G/K,χ) as a real sub-

space of Cℓ(G/K)⊗C S(G/K,χ). In this way we view grad0ψ as an element of

Cℓ(G/K)⊗C S(G/K,χ), to which we can apply κ.

Definition 5.7. Let (G/K, g0, J) be almost Hermitian, and let χ be a char-acter of K. Let ∇ be a G-invariant connection on T (G/K) that is compatiblewith g0 and commutes with J . Then the Dirac operator, D∇, for ∇ and χ isdefined on S(G/K,χ) by

D∇ψ = κ(grad0ψ).

We remind the reader that κ depends on the choice of g0 and J , and thatgrad0 ψ also depends on the choice of ∇.

In the setting of Definition 5.7 we can use a standard module frame Wjfor T (G/K) and g0 to give a more explicit description of grad0

ψ , namely

grad0ψ =

∑

j

Wj ⊗ (∇SWjψ).

(See the paragraph of [37] containing Equation 8.2.) In terms of Wj we canthen write D∇ as

(5.8) D∇ψ =∑

j

κ(Wj)(∇SWjψ).

These expressions for grad0ψ and D∇ are, of course, independent of the choice

of standard module frame. This can be seen directly by using Proposition 2.9.

Proposition 5.9. The operator D∇ commutes with the action of G on S(G/K,χ) by translation, and anti-commutes with the chirality operator κ(γ).

Proof. The commutation with the action of G is easily verified, much as donein the paragraph after Equation 8.2 of [37]. As to κ(γ), we are viewing γ as aconstant field in Cℓ(G/K), and because LX ∈ u(mJ) for each X ∈ m we haveLV γ = 0, as seen near the end of the previous section. Since γ is constant, we



also clearly have ∇cV γ = 0, and thus ∇V γ = 0 for all V ∈ T (G/K). Then fromProposition 5.6 we see that κ(γ) commutes with ∇SV for each V ∈ T (G/K).Since γ anti-commutes with each X ∈ m ⊂ Cℓ(m), it follows easily that D∇

anti-commutes with κ(γ).

We have been viewing S(G/K,χ) as a right AC-module. But since AC iscommutative we can equally well view S(G/K,χ) as a left AC-module, andthis view is quite natural when we view AC as the center of Cℓ(G/K). For anyf ∈ AC we let Mf denote the operator on S(G/K,χ) consisting of pointwisemultiplication by f , viewed as acting on the left of S(G/K,χ). By means of g0we can identify df (defined by df(V ) = δV (f)) with an element of TC(G/K),

which we denote by grad0f since it is the usual gradient of f for g0. In terms

of a standard module frame, Wj, for T (G/K) we have

grad0f =

∑

j

(δWjf)Wj ,

with the evident meaning considering that δWjf is C-valued. Then it is easily

seen, much as in the proof of Proposition 8.3 of [37], that:

Proposition 5.10. For any f ∈ AC and ψ ∈ S(G/K,χ) we have

[D∇,Mf ]ψ = κ(grad0f )(ψ).

For the reader’s convenience we now basically repeat the comments maderight after the proof of Theorem 8.4 of [37]. Let the Hilbert space L2(G/K,S)be defined in terms of the G-invariant measure on G/K from that on G. Bychoosing a fundamental domain in G we can view S(G/K,χ) as a dense sub-space of L2(G/K,S). In this way D∇ can be viewed as an unbounded op-erator on L2(G/K,S). Note that for different choices of χ the spectrum ofD∇ can be quite different. We equip S(G/K,χ) with the inner product fromL2(G/K,S), which will just be

∫G〈ψ, ϕ〉AC

. For f ∈ AC we let Mf denote also

the corresponding operator on L2(G/K,S) by pointwise multiplication. FromProposition 5.10 we see that the operator norm of the commutator [D∇,Mf ] is

the same as that of κ(grad0f ) as an operator on S(G/K). Recall from Equation

4.9 that κ is a ∗-representation. For any c ∈ Cℓ(G/K) let ‖κ(c)‖ denote theoperator norm of κ(c) as an operator on S(G/K,χ). Then by the C∗-identity‖T ‖2 = ‖T ∗T ‖ we see that ‖κ(c)‖2 = ‖κ(c∗c)‖. When c = V ∈ T (G/K) thismeans that

‖κ(V )‖2 = ‖κ(〈V, V 〉A)‖ = ‖M〈V,V 〉A‖ = ‖〈V, V 〉A‖∞ = ‖V ‖2∞,for the evident meaning of the last term, where ‖·‖∞ is just the usual supremumnorm. Notice that this is independent of the choice of χ (basically reflectingthe fact that the C∗-norm on a full matrix algebra is unique). When we applythis for V = grad0

f we obtain

‖κ(grad0f )‖ = ‖ grad0

f ‖∞.


288 Marc A. Rieffel

Now a standard argument (e.g., following Definition 9.13 of [19]) shows that ifwe denote by ρ the ordinary metric on a Riemannian manifold N coming fromits Riemannian metric, then for any two points p and q of N we have

ρ(p, q) = sup|f(p)− f(q)| : ‖ gradf ‖∞ ≤ 1.On applying this to G/K and using Proposition 5.10 and the discussion fol-lowing its proof, we obtain, for ρ now the ordinary metric on G/K from ourRiemannian metric g0,

ρ(p, q) = sup|f(p)− f(q)| : ‖[D∇,Mf ]‖ ≤ 1.This is the formula on which Connes focused for general Riemannian manifolds[12, 13], as it shows that the Dirac operator contains all of the metric informa-tion (and much more) for the manifold. This is his motivation for advocatingthat metric data for “noncommutative spaces” be encoded by providing themwith a “Dirac operator”. But we should notice that our Dirac operators abovemay not be formally self-adjoint. We deal with that issue in the next section.

We remark that the first part of Proposition 5.9 is the manifestation interms of D∇ of the fact that the ordinary metric on G/K for g0 is invariantfor the action of G on G/K.

At this point it is clear that we can combine the construction of this sectionwith the formula in Theorem 3.3 for the Levi-Civita connection for a coadjointorbit to obtain a fairly explicit formula for the canonical Dirac operator forthe coadjoint orbit of µ ∈ g′. But we refrain from writing this formula here asit is somewhat lengthy, and we do not need it for the next section.

6. The formal self-adjointness of the Dirac operator

By definition, D∇ will be formally self-adjoint if

〈D∇ψ, ϕ〉 = 〈ψ,D∇ϕ〉for any ϕ, ψ ∈ S(G/K,χ), where the inner product is that from L2(G/K,S).Recall that the torsion, T∇, of a connection ∇ on T (G/K) is defined by

T∇(V,W ) = ∇VW −∇WV − [V,W ]

for V,W ∈ T (G/K). Note that [V,W ] is defined as the commutator of deriva-tions of A, and that when elements of T (G/K) are viewed as functions as wehave been doing, then [V,W ] is not defined pointwise, but rather has a some-what complicated expression in terms of V and W . But in Section 6 of [37] itis seen that the function [V,W ] can be readily calculated when V and W arefundamental vector fields, and we will use this fact later. It is not difficult tosee that T∇ is A-bilinear. (See §8 of Chapter 1 of [22].) For any U ∈ T (G/K)let TU∇ be the A-endomorphism of T (G/K) defined by TU∇ (V ) = T∇(U, V ). Wecan define trace(TU∇ ) by trace(TU∇ ) =

∑j g0(T∇(U,Wj),Wj) for one (hence ev-

ery, by Proposition 2.9) standard module frame Wj for T (G/K) equippedwith g0.



The purpose of this section is to prove the following theorem, and to obtainsome of its consequences. As we will see, this theorem is closely related to themain theorem of [23], which deals with the case in which G/K is spin. (Seealso [17].)

Theorem 6.1. Let (G/K, g0, J) be almost Hermitian, and let χ be a characterof K. Let ∇ be a G-invariant connection on T (G/K) that is compatible withg0 and commutes with J , so that we can define the Dirac operator D∇ onL2(G/K,S), with domain S(G/K,χ), as explained in the previous section.Then D∇ is formally self-adjoint if and only if

trace(TU∇ ) = 0

for every U ∈ T (G/K).

Proof. We try to follow the path of the proof of Theorem 8.4 in the latestrevised arXiv version of [37]. (The published version of this paper has a seriouserror in the proof of Theorem 8.4, and that error is corrected in the mostrecent arXiv version.) We use first the Leibniz rule of Proposition 5.6 andthen the compatibility of ∇S with the inner product to calculate that forψ, ϕ ∈ S(G/K,χ) we have

(6.2)

〈D∇ψ, ϕ〉AC− 〈ψ,D∇ϕ〉AC

=∑

j

(〈κ(Wj)(∇SWjψ), ϕ〉AC

− 〈ψ, κ(Wj)(∇SWjϕ)〉AC

)

=∑

j

(−〈∇SWjψ, κ(Wj)ϕ〉AC

− 〈ψ,∇SWj(κ(Wj)ϕ)

− κ(∇WjWj)ϕ〉AC

)

= −∑

j

δWj(〈ψ, κ(Wj)ϕ〉AC

) + 〈ψ, κ(∑

j

∇WjWj)ϕ〉AC

.

For given ψ and ϕ the function V 7→ 〈ψ, κ(V )ϕ〉ACfor V ∈ T (G/K) is A-

linear. It is not C-linear for the complex structure on T (G/K) from J , be-cause κ is not C-linear, as was mentioned immediately after the definition ofκ = κJ in Section 4. Of course, the above function does extend to an AC-linear function from the complexification, TC(G/K), of T (G/K). We equipTC(G/K) with the complexification of the inner product on T (G/K) from g0.Clearly TC(G/K) corresponds to the “induced bundle” for the Ad-action of Kon the complexification, mC, of m. Every AC-linear function from TC(G/K)into AC is represented through the inner product by an element of TC(G/K).(See, e.g., Proposition 7.2 of [36].) Thus there is a U ∈ TC(G/K) such that〈ψ, κ(V )ϕ〉AC

= 〈U, V 〉ACfor all V ∈ T (G/K).

Lemma 6.3. Let E denote the C-linear span of the U ’s that arise as abovefrom pairs (ψ, φ) of elements of S(G/K,χ). Then E = TC(G/H).

Proof. It suffices to show that T (G/K) is in E . So let U ∈ T (G/K) begiven. Let also a cross-section h for the line-bundle for χ be given, so that


290 Marc A. Rieffel

h ∈ C∞(G,C) and h(xs) = χ(s)h(x) for all x ∈ G and s ∈ K. Let 1S denotethe identity element of S = F(mJ), and let φ be defined by φ(x) = h(x)1S .View U as having values in mJ , and let ψ be defined by ψ(x) = h(x)U(x),using J to define the C-space structure of mJ . Then both ψ and φ are inS(G/K,χ). For any V ∈ T (G/K) we then have

(κ(V )φ)(x) = ia†J(V (x))(h(x)1S ) = ih(x)V (x).

Thus, with 〈·, ·〉J defined on mJ as done in Section 4 and on S as done inEquation 4.3, we have

〈ψ, κ(V )φ〉AC(x) = i|h(x)|2〈U(x), V (x)〉J

= i|h(x)|2(g0(U(x), V (x)) + ig0(JU(x), V (x))).

But because κ(V ) is a skew-adjoint operator on S(G/K,χ), we have

〈ψ, κ(V )φ〉−AC= 〈κ(V )φ, ψ〉AC

= −〈φ, κ(V )ψ〉AC.

Thus the real and imaginary parts of the function V 7→ 〈ψ, κ(V )φ〉ACare both

in E , and so that the function V 7→ |h|2g0(U, V ) is in E . Note that |h|2 ∈ A.Now let hj be a standard module frame for the line-bundle for χ, so that∑ |hj |2 = 1. Since each function V 7→ |hj |2g0(U, V ) is in E , by summing themover j we see that the function V 7→ g0(U, V ) is in E , as desired.

Now in terms of the U for ψ and ϕ the expression (6.2) becomes:

= −∑

j

δWj(〈U,Wj〉AC

) + 〈U,∑

j

∇WjWj〉AC

) =∑

j

〈∇WjU,Wj〉AC

.

Thus from Lemma 6.3 we see that D∇ is formally self-adjoint if and only if∫

G/K

∑

j

〈∇WjU,Wj〉AC

= 0

for all U ∈ TC(G/K). By expressing the real and imaginary parts of the innerproduct in terms of g0, and expressing U in terms of its real and imaginaryparts, we see that we have obtained:

Lemma 6.4. With notation as above, the Dirac operator D∇ is formally self-adjoint if and only if ∫

G/K

∑

j

g0(∇WjU,Wj) = 0

for all U ∈ T (G/K) and one (hence every) standard module frame, Wj, forT (G/K) and g0.

Thus we have reduced the matter to a condition concerning ∇ on T (G/K),so J is no longer involved, we no longer need to consider the Clifford algebraand spinors, and we can work over R from this point on. From the definitionof the torsion, T∇, of ∇ we have

g0(∇WjU,Wj) = g0(∇UWj − T∇(U,Wj)− [U,Wj ],Wj)



for each j. Notice now that∑j g0(Wj ,Wj) is independent of the choice of

standard module frame, by the A-bilinearity of g0 and by Proposition 2.9. Forany given x ∈ G we have

∑

j

(g0(Wj ,Wj))(x) =∑

j

g0(Wj(x),Wj(x)),

and Wj(x) forms a frame for m with g0. By Proposition 2.9 the expressionon the right is independent of the choice of frame for m, and so we can use anorthonormal basis for m and g0. From this we see that

∑

j

g0(Wj ,Wj) ≡ dim(m).

Consequently, by the compatibility of ∇ with g0, for any U ∈ T (G/K) we have

0 = δU (∑

j

g0(Wj ,Wj)) =∑

j

g0(∇UWj ,Wj) + g0(Wj ,∇UWj)

= 2∑

j

g0(∇UWj ,Wj).

Thus ∑

j

g0(∇UWj ,Wj) = 0.

(We remark that this fact depends on the pointwise argument used just above,and that the analogous argument can fail for modules over a noncommutativeA that contains proper isometries.) We see thus that

∑

j

g0(∇WjU,Wj) = −

∑

j

g0(T∇(U,Wj),Wj)−∑

j

g0([U,Wj ],Wj).

Let ∇t be the Levi-Civita connection for g0. We can apply the above equa-tion to ∇t and use the fact that ∇t is torsion-free to get an expression for thelast term above. In this way we find that

∑

j

g0(∇WjU,Wj) = −

∑

j

g0(T∇(U,Wj),Wj) +∑

g0(∇tWjU,Wj).

Because ∇t is the Levi-Civita connection for g0, the sum∑

j g0(∇tWjU,Wj) is

exactly div(U) as defined in Definition 2.7. From the divergence theorem thatwas proved in Theorem 2.10 we have

∫

G/K

∑

j

g0(∇tWjU,Wj) = 0

for all U ∈ T (G/K). Thus we see that D∇ is formally self-adjoint exactly if∫

G/K

∑

j

g0(T∇(U,Wj),Wj) = 0


292 Marc A. Rieffel

for all U ∈ T (G/K). But the integrand is clearly A-linear in U , so if we replaceU by Uf for any f ∈ A the f comes outside the inner product and the sum.Since f is arbitrary, this means that the integral is always 0 exactly if

∑

j

g0(T∇(U,Wj),Wj) = 0

for all U ∈ T (G/K). But the left-hand side is exactly our definition oftrace(TU∇ ).

Note that the criterion in the theorem is independent of the choice of J (aslong as J commutes with ∇).

Corollary 6.5. Let µ⋄ ∈ g′ and let G/K correspond to the coadjoint orbitof µ⋄. Let g⋄ be the Riemannian metric on G/K corresponding to the Kahlerstructure from µ⋄, and let ∇⋄ be its Levi-Civita connection. Let D⋄ be the Diracoperator for ∇⋄ constructed as in the previous section (since ∇⋄ commutes withJ⋄), for any character of K. Then D⋄ is formally self-adjoint.

Proof. Since the torsion of ∇⋄ is 0 by definition, application of Theorem 6.1immediately shows that D⋄ is formally self-adjoint.

For any almost-Hermitian (G/K, g0, J) there is always at least one con-nection that satisfies the hypotheses of Theorem 6.1, namely the canonicalconnection ∇c. Even though it may not be torsion-free, we have:

Corollary 6.6. Let (G/K, g0, J) be almost Hermitian, and let ∇c be the canon-ical connection on T (G/K) . Let D∇

c

be the Dirac operator constructed asin the previous section for g0 using J (since ∇c is compatible with g0 andcommutes with J), for any character of K. Then D∇

c

is formally self-adjoint.

Proof. From Section 6 of [37] (where the canonical connection is denoted by∇0) we find that

(T∇c(V,W ))(x) = −P [V (x),W (x)].

Thus to apply the criterion of Theorem 6.1 we need to show that

(6.6) trace(TU∇c)(x) =∑

j

g0(P [U(x),Wj(x)],Wj(x)) = 0

for each U ∈ T (G/K) and x ∈ G. Now Wj(x) is a frame for m with respectto g0 for each x, and by Proposition 2.9 for a given x we can replace Wj(x)by an orthonormal basis for m with respect to g0. We see in this way that fora given x, if we set Y = U(x), then expression (6.6) is simply trace(P adY )where P adY is viewed as an operator on m. But the trace of an operatoris independent of any choice of inner product on the vector space. Thus wecan instead use a basis, Xj, for m that is orthonormal for Kil. Since P isself-adjoint for Kil on g, the expression (6.6) (for the given x) is just

∑Kil([Y,Xj ], Xj).



But adY is skew-adjoint for Kil on g, and so each term in the sum is 0. Thusthe criterion of Theorem 6.1 is fulfilled.

Suppose now that (G/K, g0, J) is almost Hermitian and that ∇ is a G-invariant connection on T (G/K) that is compatible with g0 and commuteswith J . As done earlier, set L = ∇ − ∇c. Then a simple calculation showsthat

T∇(V,W ) = T∇c(V,W ) + LVW − LWV,for V,W ∈ T (G/K), and so for any U ∈ T (G/K) we have

trace(TU∇ ) = trace(TU∇c) +∑

j

g0(LUWj − LWjU,Wj).

But in the proof of corollary 6.6 we verified Equation 6.6, which says thattrace(TU∇c) = 0. Furthermore, LU is skew-symmetric, so g0(LUWj ,Wj) = 0for each j. It follows that

trace(TU∇ ) = −∑

j

g0(LWjU,Wj) =

∑

j

g0(U,LWjWj).

Since we need this to be 0 for all U , we obtain:

Corollary 6.7. Let (G/K, g0, J) be almost Hermitian, and let ∇ be a G-invariant connection on T (G/K) that is compatible with g0 and commuteswith J . Let D∇ be the Dirac operator for g0 and J , for a character χ of K.Let L = ∇−∇c. Then D∇ is formally self-adjoint if and only if

∑

j

LWjWj = 0

for one, hence every, standard module frame for T (G/K) and g0.

The next results are motivated by the corollary in [23].

Lemma 6.8. For L as above,∑j LWj

Wj is a constant function on G, whosevalue is in the subspace of m consisting of elements that are invariant underthe Ad-action of K on m.

Proof. Because ∇ and ∇c are G-invariant, so is L, where this means thatλx(LWV ) = LλxW (λxV ), as seen in Section 5 of [37]. Consequently for anyx ∈ G

(∑

j

LWjWj)(x

−1) = (∑

LλxWj(λxWj))(e),

where e is the identity element of G. But λxWj is again a standard moduleframe, and the expression is independent of the choice of standard moduleframe by Proposition 2.9, so the first statement is verified. For any x ∈ G and


294 Marc A. Rieffel

s ∈ K we have

Ads((∑

LWj)(x)) = Ads(

∑LWj

(x)(Wj(x))

=∑

(Ads LWj(x) Ad−1

s )(Ads(Wj(x)))

=∑

LAds(Wj(x))(Ads(Wj(x)),

where we have used Proposition 3.1 of [37]. But again the independence of thechoice of frame shows the invariance under the Ad-action of K.

Corollary 6.9. Let G/K be the coadjoint orbit for µ⋄ ∈ g′, and let ∇ be any G-invariant connection on T (G/K) that is compatible with g⋄ and commutes withJ⋄. Then for any character χ of K the Dirac operator D∇ on S(G/K, g⋄, χ)is formally self-adjoint.

Proof. Because K contains a maximal torus, the only element of m that isinvariant for the Ad-action of K is 0.

We remark that when G/K can be identified with a coadjoint orbit, thereare usually many different coadjoint orbits to which it can be identified, andthus many different complex structures J (and Riemannian metrics) that canbe used when applying the above corollary.

From Lemma 6.8 we see that the criterion of Corollary 6.7 will be satisfiedif and only if (

∑j LWj

Wj)(e) = 0. Let Yp be a g0-orthonormal basis for m.

Then S1/2Yp will be a Kil-orthonormal basis for m, which we can extend

to a Kil-orthonormal basis Xj for g. Then Xj is a standard module Kil-

biframe for T (G/K), and so, as seen just before Theorem 2.10, (Xj, S−1Xj)

is a standard module g0-frame for T (G/K). By Proposition 2.9 the criterion

is equivalent to 0 =∑

j LXj (e)(S−1Xj(e)) Now as seen before Theorem 2.10,

X(x) = −P Ad−1x (X) for any X ∈ g, so that X(e) = −PX . Consequently

S−1Xp(e) = −S−1/2Yp for each p. In this way we obtain the following corollary,which is very similar to the criterion that Ikeda obtained for the spin case inthe main theorem of [23]:

Corollary 6.10. Let (G/K, g0, J) be almost Hermitian, and let ∇ be a G-invariant connection on T (G/K) that is compatible with g0 and commutes withJ . Let D∇ be the Dirac operator for g0 and J on S(G/K,χ) for a characterχ of K. Let L = ∇−∇c. Then D∇ is formally self-adjoint if and only if forone (and so for any) g0-orthonormal basis Yp for m we have

∑

p

LXp(e)(S−1/2Yp) = 0

where Xp = S1/2Yp for each p.

For the essential self-adjointness of Dirac operators see, for example, Sec-tion 9.4 of [19] and Section 4.1 of [16].



I thank John Lott and Mattai Varghese for independently bringing to myattention the paper [7]. In this paper connections which have nonzero torsionare considered, and in Definition 1.9 certain modified Dirac operators are de-fined, and in Theorem 1.10 these modified Dirac operators are shown to beself-adjoint. We can in the same way define self-adjoint modified Dirac opera-tors. Within the setting of Theorem 6.1 let ∇ = ∇c +L as before. Then fromthe definition of Dirac operators in terms of standard module frames givenafter Definition 5.7 we see that

D∇ψ = D∇c

ψ +∑

j

κ(Wj)LWjψ.

Let M be the operator defined by Mψ =∑

j κ(Wj)LWjψ. It is clearly a

bounded operator on L2(G/K,S). Then on S(G/K,χ) we have (D∇)∗ =(D∇

c

)∗ +M∗. But we saw in Corollary 6.6 that D∇c

is formally self-adjoint.From this we see that D∇− (D∇)∗ = M −M∗. Consequently, if we define the

modified Dirac operator by D∇ = D∇ − (1/2)(M −M∗), then it is easily seen

that D∇ is formally self-adjoint.I also thank John Lott for bringing to my attention the paper [18]. It

assumes only that K is a connected subgroup of G, and deals only with metricson G/K that are “normal”, that is come from G-invariant metrics on g. Theconnections that are considered, which can have nonzero torsion, are quitesimilar to those used in [3]. In the first two sections G/K is assumed to bespin, and the Dirac operators are self-adjoint, for reasons that appear to beclosely related to Corollary 6.6. In the next sections of [18] G/K is not assumedto be spin, but this is dealt with by tensoring the spinor representation of Cℓ(m)by suitable unitary representations of K. This appears to be related to the“spinK” structures of [5], but I have not explored this technique.

It would be interesting to know how all of the results of our paper relateto Connes’ action principle for finding the Dirac operator from among all ofthe spectral triples that give a specified Riemannian metric [13]. (See alsoTheorem 11.2 and Section 11.4 of [19].) Of course, on the face of it Connes’theorem is for spin manifolds while many homogeneous spaces are not spin.

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Received December 16, 2008; accepted April 16, 2009

Marc A. RieffelDepartment of MathematicsUniversity of California, Berkeley, CA 94720-3840, USAE-mail: [email protected]




Duality of cones of positive maps

Erling Størmer



Abstract. We study the so-called K-positive linear maps from B(L) into B(H) for finitedimensional Hilbert spaces L and H corresponding to a mapping cone K and give char-acterizations of the dual cone of the cone of K-positive maps. Applications are given todecomposable maps and their relation to PPT-states.

Introduction

The study of positive linear maps of C*-algebras, and in particular thoseof finite dimensions, has over the last years been invigorated by its connectionwith quantum information theory. While most work on positive maps on C*-algebras has been related to completely positive maps, in quantum informationtheory other classes of maps appear naturally. In [6] the author introduceddifferent cones of positive maps and defined what he called K-positive mapsarising from a so-called mapping cone K of positive maps of B(H) into itself,not to be confused with k-positive maps for k ∈ N. see section 2 for detailsof this and the following. In [1, Section 11.2], the authors introduced whatthey called the dual cone of a cone of positive maps, see also [9]. In thepresent paper we shall follow up this idea by studying the dual cone of thecone of K−positive maps. Our main result gives several characterizations ofwhen a map belongs to a dual cone; in particular the result is an extensionof the Horodecki Theorem [3] to general mapping cones. Then we show thatthe dual of a dual cone equals the original cone, and if the mapping cone Kis invariant under the action of the transpose map, then for maps of B(H)into itself the dual cone consists of K♯-positive maps for a mapping cone K♯

naturally defined by K. Applications are given to the most studied maps, likecompletely positive, copositive, and decomposable maps, and to maps definedby separable states and PPT-states. In particular it is shown that if P is themapping cone of maps which are both completely positive and copositive, i.e.

300 Erling Størmer

maps which correspond to PPT-states, then the P−positive maps constituteexactly the dual of the cone of decomposable maps.

1. Dual cones

In this section we shall study certain cones of positive maps from the com-plex n × n matrices Mn, denoted by M below, to the bounded operatorsB(H) on a Hilbert space H, which we for simplicity assume is finite dimen-sional. We denote by B(M,H) (resp. B(M,H)+, CP (M,H), Cop(M,H)) thelinear (resp. positive, completely positive, and copositive) maps of M intoB(H). Recall that a map φ is copositive if t φ is completely positive, t beingthe transpose map. When M = B(H) we shall use the simplified notationP(H) = B(B(H), H)+. Recall from [6] that a mapping cone is a nonzeroclosed cone K ⊂ P(H) such that if φ ∈ K and a, b ∈ B(H) then the mapx → aφ(bxb∗)a∗ belongs to K. Since every completely positive map in P(H)is of the sum of maps x → axa∗, it follows that if φ ∈ K then α φ β ∈ Kfor all α, β ∈ CP (H) - the completely positive maps in P(H). We also denoteby Cop(H) the copositive maps in P(H). Let

P (M,K) = x ∈M ⊗B(H) : ι⊗ α(x) ≥ 0, ∀α ∈ K,where ι denotes the identity map. By [6, Lemma 2.8], P (M,K) is a properclosed cone. If Tr denotes the usual trace on B(H), and also on M andM ⊗ B(H) when there is no confusion of which algebra we refer to, and φ ∈B(M,H)+, then the dual functional φ on M ⊗B(H) is defined by

φ(a⊗ b) = Tr(φ(a)bt).

We say φ is K-positive if φ is positive on the cone P (M,K). It was shown in[6] that if K = CP (H), then φ is K−positive if and only if φ is completelypositive. Other characterizations will be shown below. We denote by PK(M)the cone of K−positive maps.

If (eij) is a complete set of matrix units in M we denote by p the rank 1operator p =

∑ij eij ⊗ eij ∈ M ⊗M, and if φ ∈ B(M,H) we let Cφ denote

the Choi matrix

Cφ = ι⊗ φ(p) =∑

ij

eij ⊗ φ(eij) ∈M ⊗B(H).

Then φ is completely positive if and only if Cφ ≥ 0, [2]. We shall showcharacterizations of other classes of maps by positivity properties of Cφ. IfS ⊂ B(M,H)+ then its dual cone is defined by

S = φ ∈ B(M,H) : Tr(CφCψ) ≥ 0, ∀ψ ∈ S.If φ ∈ B(M,H) we denote by φt the map t φ t. Then Cφt is the density

operator for φ, i.e. φ(x) = Tr(Cφtx), see [7, Lemma 5]. If K is a mappingcone we put

Kt = φt : φ ∈ K.Then Kt is also a mapping cone, and is in many cases equal to K.


Duality of cones of positive maps 301

We denote by φ∗ the adjoint map of φ considered as a linear operator ofB(H) into M associated with the Hilbert-Schmidt structure, viz.

Tr(φ(a)b) = Tr(aφ∗(b)), a ∈M, b ∈ B(H).

We can now state the main result of this section.

Theorem 1. Let H be a finite dimensional Hilbert space and K a mappingcone in P(H). Let PK(M) be the dual cone of the K−positive maps PK(M).Let φ ∈ B(M,H). Then the following conditions are equivalent.

(i) φ ∈ PK(M).

(ii) Cφ ∈ P (M,Kt).

(iii) φ (ι⊗ α∗) ≥ 0, ∀α ∈ K.

(iv) α φ ∈ CP (M,H), ∀α ∈ Kt.

In [8] we showed a version of the Horodecki Theorem [3] which in the casewhen n ≤ dimH is equivalent to the Horodecki Theorem. We obtain this andmore as a corollary to Theorem 1, thus showing that it can be viewed as anextension of the Horodecki Theorem to arbitrary mapping cones, see also [4].Note that P(H) is a mapping cone containing all others, see [6, Lemma 2.4].

Corollary 2. Let φ ∈ B(M,H)+. Then the following four conditions areequivalent.

(i) φ ∈ PP(H)(M).

(ii) ι⊗ α(Cφ) ≥ 0, ∀α ∈ P(H).

(iii) φ (ι⊗ α) ≥ 0, ∀α ∈ P(H).

(iv) α φ ∈ CP (M,H), ∀α ∈ P(H).Furthermore, if n ≤ dimH then the above conditions are equivalent to

(v) φ is separable.

Proof. Since P(H) = P(H)t the four conditions, (i)-(iv) are equivalent by thecorresponding conditions in Theorem 1. By [8, Lemma 9], (iii)⇔ (v) whenn ≤ dimH , proving the last part of the corollary.

For the proof of the theorem we shall need some lemmas. The first caneasily be extended to the general situation studied in [6].

Lemma 3. Let ρ be a linear functional on M ⊗ B(H) with density operatorh. Let K be a mapping cone in P(H). Then h ∈ P (M,K) if and only ifρ (ι⊗ α∗) ≥ 0, ∀α ∈ K.

Proof. ρ(ι⊗α∗)(x) = Tr(h(ι⊗α∗)(x)) = Tr(ι⊗α(h)x). Hence ρ(ι⊗α∗) ≥ 0for all α ∈ K if and only if ι⊗α(h) ≥ 0 for all α ∈ K if and only if h ∈ P (M,K),completing the proof.

Recall that if φ ∈ B(M,H)+ then φt(x) = φ(xt)t = t φ t(x).


302 Erling Størmer

Lemma 4. Let φ ∈ B(M,H)+ Then

(i) φt = φ (t⊗ t).(ii) Cφt = t⊗ t(Cφ) = Ctφ.

(iii) φt(x) = φ(xt).

Proof. (i) By definition of φt and φ we have

φt(a⊗ b) = Tr(φt(a)bt) = Tr(φ(at)btt) = φ(at ⊗ bt) = φ (t⊗ t)(a⊗ b).(ii) By (i) and the fact that Cφ is the density operator for φt, we have

Tr(Cφa⊗ b) = φt(a⊗ b) = Tr(φt(a)bt) = Tr(φ(at)tbt)

= φ(at ⊗ bt) = Tr(Cφtat ⊗ bt) = Tr(t⊗ t(Cφt)a⊗ b).

Hence Cφ = t⊗ t(Cφt), and (ii) follows.(iii) By (ii)

φ(xt) = Tr(Cφtxt) = Tr(Ctφx

t) = Tr(Cφx) = φt(x),

completing the proof.

Lemma 5. If K is a mapping cone, then

P (M,Kt) = Cφ : Ctφ ∈ P (M,K) = t⊗ t(P (M,K)).

Proof. We have

ι⊗ αt(x) = (ι⊗ t) (ι ⊗ α) (ι⊗ t)(x) = (t⊗ t) (ι⊗ α) (t⊗ t)(x).Thus x ∈ P (M,Kt) if and only if t ⊗ t(x) ∈ P (M,K), if and only if x ∈t⊗ t(P (M,K)), and the two cones are equal.

Each operator x ∈M⊗B(H) is of the form Cφ for some map φ ∈ B(M,H).By Lemma 4(ii) and the above we therefore have that Cφt = t ⊗ t(Cφ) ∈P (M,K) if and only if Cφ ∈ t ⊗ t(P (M,K)) = P (M,Kt), completing theproof.

Proof of Theorem 1. (i)⇔(ii) As before let p =∑

ij eij ⊗ eij , where (eij) is a

complete set of matrix units for M = Mn. Then Cφ = ι⊗φ(p). By [6, Theorem3.6], the cone of K−positive maps PK(M) is generated as a cone by maps ofthe form α ψ with α ∈ Kd = t α∗ t : α ∈ K, and ψ ∈ CP (M,H). Wethus have

φ ∈ PK(M) ⇔ Tr(CφCαψ) ≥ 0, ∀α ∈ Kd, ψ ∈ CP (M,H),

hence if and only if

0 ≤ Tr(Cφ(ι⊗ α)(Cψ)) = Tr(ι⊗ α∗(Cφ)Cψ), ∀α, ψas above, which holds if and only if ι ⊗ α∗(Cφ) ≥ 0, ∀α ∈ Kd, since (M ⊗B(H))+ = Cψ : Cψ ∈ CP (M,H). Thus φ ∈ PK(M) if and only if ι ⊗αt(Cφ) ≥ 0, ∀α ∈ K if and only if Cφ ∈ P (M,Kt).



(i)⇔(iii) By Lemma 5 Cφ ∈ P (M,Kt) if and only if Cφt ∈ P (M,K). Hence

by (i)⇔(ii) φ ∈ PK(M) if and only if Cφt ∈ P (M,K), hence by Lemma 3, if

and only if φ (ι⊗ α∗) ≥ 0, ∀α ∈ K, proving (i)⇔(iii).(ii)⇔(iv) Recalling the definitions of Cφ, P (M,K) and their properties we

have

Cφ ∈ P (M,Kt)⇔ Cαtφ = ι⊗ αt(Cφ) ≥ 0, ∀α ∈ K⇔ β φ ∈ CP (M,H), ∀β ∈ Kt.

This completes the proof of the theorem.

We conclude the section by showing that taking the dual is a well behavedproperty.

Theorem 6. Let K be a mapping cone. Then (PK(M)) = PK(M).

Proof. Let φ ∈ B(M,H)+. Then

φ ∈ (PK(M)) ⇔ Tr(CφCψ) ≥ 0, ∀ψ ∈ PK(M)

⇔ Tr(CφCψ) ≥ 0, ∀Cψ ∈ P (M,Kt) by Thm. 1

⇔ Tr(Cφ(t⊗ t)(Cρ)), ∀Cρ ∈ P (M,K) by Lem. 5

⇔ Tr(CtφCρ) ≥ 0, ∀Cρ ∈ P (M,K)

⇔ φ(Cρ) ≥ 0, ∀Cρ ∈ P (M,K) by Lem.4(ii)

⇔ φ ∈ PK(M).

The proof is complete.

In the notation of [6] S(H) denotes the mapping cone consisting of mapsof the form x 7→ ∑

ωi(x)bi, with ωi states of B(H) and bi ∈ B(H)+. By [7,

Theorem 2], φ is S(H)−positive if and only if φ is separable. It follows from

Lemma 2.1 in [6] that φ is positive if and only if φ is positive on the coneB(H)+ ⊗B(H)+ generated by operators of the form a⊗ b with a, b ∈ B(H)+,which holds if and only if Tr(Cφx) ≥ 0 for all x ∈ B(H)+ ⊗ B(H)+. But bythe above P (B(H), P (H)) = B(H)+ ⊗ B(H)+. Thus φ ∈ P(H) if and onlyif φ ∈ PS(H)(B(H)), and by Theorem 6 φ is S(H)−positive if and only if

φ ∈ P(H).

2. Maps on B(H)

In the previous section we gave characterizations of the dual cone PK(M)

of a mapping cone K. A natural question is whether PK(M) equals the conePK♯(M) for some mapping cone K♯. In the present section we shall do this formaps in P(H) = B(B(H), H)+ when K is a mapping cone invariant under thetranspose map, viz. K = Kt. The cone K♯ is defined in our first lemma.

Lemma 7. Let K be a mapping cone. Let CK denote the closed cone generatedby all cones

ι⊗ α∗((M ⊗B(H))+), α ∈ K.


304 Erling Størmer

Let

K♯ = β ∈ P(H) : ι⊗ β(x) ≥ 0, ∀x ∈ CK.Then K♯ is a mapping cone, and furthermore

K♯ = β ∈ P(H) : β α∗ ∈ CP (H), ∀α ∈ K.Proof. Let β ∈ K♯ and γ ∈ CP (H). Then clearly γ β ∈ K♯. If α ∈ K andγ ∈ CP (H), then

(β γ) α∗ = β (γ α∗) = β (α γ∗)∗.Since K is a mapping cone, and γ ∈ CP (H), αγ∗ ∈ K, hence ι⊗β(αγ∗)∗ ≥0, so that β γ ∈ K♯, proving the first part of the lemma. To show the secondpart we have that β ∈ K♯ if and only if

ι⊗ β α∗(x) = ι⊗ β(ι ⊗ α∗)(x) ≥ 0, ∀x ≥ 0, α ∈ K,which holds if and only if β α∗ ∈ CP (H), because a map γ ∈ P(H) iscompletely positive if and only if ι ⊗ γ ≥ 0 on B(H ⊗ H)+. The proof iscomplete.

We shall need the following rephrasing of Choi’s result [2] that φ is com-pletely positive if and only if Cφ ≥ 0.

Lemma 8. Let φ ∈ P(H) and ω the maximally entangled state, viz. ω(x) =1nTr(px), n = dimH. Then φ ∈ CP (H) if and only if ω (ι⊗ φ) ≥ 0.

Proof. φ is completely positive if and only if

0 ≤ Tr(Cφx) = Tr(p(ι⊗ φ∗)(x)) = nω (ι⊗ φ∗)(x), ∀x ≥ 0.

Since CP (H) is closed under the *-operation, the lemma follows.

Lemma 9. Let K be a mapping cone and CK as in Lemma 7. Let φ ∈ P(H).Then we have

(i) φ is positive on CK if and only if φ∗t ∈ K♯.

(ii) φt is positive on CK if and only if φ∗ ∈ K♯.

If K = Kt then φ is positive on CK if and only if φt is positive on CK .

Proof. (i) Let α ∈ K. Then

φ(ι⊗ α∗(x)) = Tr(Cφt(ι⊗ α∗)(x))= Tr(ι⊗ φt(p)(ι⊗ α∗)(x))= Tr(p(ι⊗ (φt∗ α∗))(x))= Tr(p(ι⊗ (φ∗t α∗))(x)),

since φt∗ = φ∗t. By Lemma 8 it follows that φ≥ 0 on CK if and only if φ∗tα∗ ∈CP (H), hence if and only if φ∗t ∈ K♯, by Lemma 7.

(ii) follows from (i) by applying (i) to φt.Assume K = Kt. If φ∗t ∈ K♯ then by Lemma 7 φ∗t α∗ ∈ CP (H) for all

α ∈ K, hence by assumption for all α∗t, α ∈ K. Thus (φ∗ α∗)t = φ∗t α∗t ∈



CP (H), hence φ∗α∗ ∈ CP (H) for all α ∈ K. Thus φ∗ ∈ K♯, so by (i) and (ii),

if φ ≥ 0 on CK then φt ≥ 0 on CK . Similarly we get the converse implication.The proof is complete.

Lemma 10. Let π : B(H)⊗B(H)→ B(H) be defined by π(a⊗b) = bta. Thenthe function Tr π is positive and linear. Furthermore, if φ ∈ P(H) then

φ = Tr π (ι⊗ φ∗t).Proof. Linearity is clear. To show positivity let x =

∑ai⊗bi ∈ B(H)⊗B(H).

Then

(Tr π)(xx∗) =∑

Tr π(aia∗j ⊗ bib∗j )

=∑

Tr(bt∗j btiaia

∗j )

=∑

Tr(a∗jbt∗j b

tiai)

= Tr((∑

btjaj)∗(∑

btiai)) ≥ 0,

so Tr π is positive. The last formula follows from the computation

φ(a⊗ b) = Tr(φ(a)bt) = Tr(aφ∗(bt)) = Tr(aφ∗t(b)t) = Tr π(ι ⊗ φ∗t(a⊗ b)).

Lemma 11. Assume that K = Kt for K a mapping cone. Then CK =P (B(H),K♯).

Proof. By definition of K♯, CK ⊂ P (B(H),K♯). Suppose y0 ∈ B(H) ⊗ B(H)and y0 is not in CK . By the Hahn-Banach Theorem there is a linear functionalφ on B(H)⊗B(H) which is positive on CK , and φ(y0) < 0. By Lemma 9 and

the assumption that K = Kt, φt ≥ 0 on CK as well, and φ∗t ∈ K♯. Write y0in the form y0 =

∑ai ⊗ bi, ai, bi ∈ B(H), and let π be as in Lemma 10. then

we have

Tr π(ι ⊗ φ∗t(y0)) = Tr π(∑

ai ⊗ φ∗(bti)t)

= Tr(∑

φ∗(bti)ai)

= Tr(∑

btiφ(ai)) = φ(y0) < 0.

Since by Lemma 10, Tr π is positive, ι ⊗ φ∗t(y0) is not a positive operator,hence y0 does not belong to P (B(H),K♯), hence CK = P (B(H),K♯). Theproof is complete.

We can now prove the main result in this section, which shows that everymap in PK(B(H)) is K♯−positive when K = Kt. Thus Theorem 1 yieldsequivalent conditions for K−positivity when K = Kt, by replacing K♯ by K,

Theorem 12. Let K be a mapping cone such that K = Kt. We then havePK(B(H)) = PK♯(B(H)), so that PK(B(H)) = PK♯(B(H)).


306 Erling Størmer

Proof. By Theorem 1(iii) φ ∈ PK(B(H)) if and only if φ is positive on CK ,hence by Lemma 11 if and only if φ is K♯−positive, i.e. φ ∈ PK♯(B(H)). Thelast statement follows from Theorem 6.

3. Decomposable maps

It was shown in [3] that a state ρ on M2 ⊗M2 or M2 ⊗M3 is separable ifand only if ρ (ι ⊗ t) ≥ 0, i.e. if and only if it is a PPT state (equivalently,ρ is said to satisfy the Peres condition). They did this by using the fact thatB(M2,C2)+ and B(M2,C3)+ consist of decomposable maps, i.e. maps whichare sums of completely positive and copositive maps. We shall in this sectionshow characterizations of decomposable maps which yield characterizations ofPPT states.

Let D denote the set of α ∈ P(H) such that α is decomposable. Then Dis the closed cone generated by CP (H) and Cop(H). Let P denote the set ofα ∈ P(H) such that α is both completely positive and copositive. Then wehave

D = CP (H) ∨ Cop(H). P = CP (H) ∩Cop(H).

Theorem 13. Let M = Mn, and D and P as above. Then D and P aremapping cones and satisfy the identities

PP (M) = PD(M), PP (M) = PD(M).

Note that by [7, Proposition 4], P consists of the maps φ ∈ P(H) such that

φ is a PPT state. The proof of Theorem 13 is divided into some lemmas.

Lemma 14. In the notation of Theorem 13 let E and F denote the cones

E = x ∈M ⊗B(H) : x ≥ 0 ∨ x ∈M ⊗B(H) : ι⊗ t(x) ≥ 0.F = x ∈M ⊗B(H) : x ≥ 0, and ι⊗ t(x) ≥ 0.

Then the following conditions are equivalent for φ ∈ B(M,H)+.

(i) φ is both completely positive and copositive.

(ii) Cφ ∈ F .

(iii) φ ≥ 0 on E.

Proof. Note that E and F are closed under the action of ι⊗ t.(i)⇒(ii) Since φ is completely positive Cφ ≥ 0, and since φ is copositive

ι⊗ t(Cφ) ≥ 0. Thus Cφ ∈ F .

(ii)⇒ (i) If x ≥ 0 then φ(x) = Tr(Cφtx) ≥ 0, so φ is completely positive. Ifι⊗ t(x) ≥ 0 then

φ(x) = Tr(Cφtx) = Tr(ι⊗ t(Cφt)ι⊗ t(x)) ≥ 0,

so φ is copositive.(ii)⇒ (iii) The same argument as for (ii)⇒ (i) applies.

(iii)⇒ (ii) If x ≥ 0 then Tr(Cφtx) = φ(x) ≥ 0. Similarly, if ι⊗ t(x) ≥ 0 then

0 ≤ φ(x) = Tr(Cφtx) = Tr(ι⊗ t(Cφt)ι⊗ t(x)) ≥ 0,



hence ι⊗ t(Cφt) ≥ 0. The proof is complete.

Lemma 15. Let K and L be mapping cones in P(H) and M as before. Then

P (M,K ∨ L) = P (M,K) ∩ P (M,L).

Proof. x ∈ P (M,K∨L) if and only if ι⊗α(x) ≥ 0 for all α ∈ K∪L, if and onlyif ι⊗α(x) ≥ 0 whenever α ∈ K or α ∈ L, if and only if x ∈ P (M,K)∩P (M,L),proving the lemma.

Note that we did not use that M is the n× n matrices in the above proof,so the lemma is true for M replaced by an operator system.

Lemma 16. Let E be as in Lemma 14. Then

P (M,P ) = P (M,CP (H)) ∨ P (M,Cop(H)) = E.

Proof. Since P (M,CP (H)) = (M ⊗B(H))+ and P (M,Cop(H)) = ι⊗ t(M ⊗B(H))+), it is clear that P (M,CP (H)) ∨ P (M,Cop(H)) = E.

If K and L are mapping cones with K ⊂ L and M is a finite dimensionalC∗−algebra then clearly P (M,L) ⊂ P (M,K), so clearly P (B(H), P ) containsthe right side of the lemma. Suppose the inclusion is proper. By the Hahn-Banach Theorem there exists a linear functional φ which is positive on Eand for some x ∈ P (B(H), P ), φ(x) < 0. By Lemma 14 φ is both completelypositive and copositive, hence so is φ∗t, so that by Lemma 10

φ(x) = Tr π(ι⊗ φ∗t(x)) ≥ 0,

a contradiction. This proves equality of the cones.

Lemma 17. With the previous notation we have

P (M,D) = F = Cβ : β ∈ PP (M).PP (M) = β : Cβ ∈ F.

Proof. By [6, Theorem 3.6], PP (M) is generated by maps αψ with α ∈ P, ψ ∈CP (M,H). By Lemma 15

F = P (M,CP (H)) ∩ P (M,Cop(H)) = P (M,D).

Let γ ∈ D. Then, with α and ψ as above,

ι⊗ γ(Cαψ) = ι⊗ γ ι⊗ α(Cψ) = ι⊗ γ α(Cψ) ≥ 0,

because ψ ∈ CP (M,H), so Cψ ≥ 0, and γ α is completely positive sinceγ is decomposable and α ∈ P. Thus Cαψ ∈ P (M,D) = F. It follows thatCβ ∈ P (M,D) = F for all β ∈ PP (M). Since D = Dt, by Lemma 5

P (M,D) = Cβ : Cβ ∈ P (M,D) = Cβ : Cβ ∈ F.By Lemma 16 if β ∈ B(M,H) then β ∈ PP (M) if and only if β ≥ 0 on E, soby Lemma 14, if and only if Cβ ∈ F. Hence PP (M) = β : Cβ ∈ F, hence

P (M,D) = F = Cβ : β ∈ PP (M),as asserted. The proof is complete.


308 Erling Størmer

Proof of Theorem 13. This is immediate, since by Lemma 17 φ ∈ PP (M) if

and only if φ(Cβ) ≥ 0 for all β ∈ PP (M) if and only if φ ≥ 0 on P (M,D), i.e.φ ∈ PD(M). The other identity follows from the first and Theorem 6.

For the rest of this section we consider the case when M = B(H).

Theorem 18. Let D and P be as in Theorem 13. Then D = P ♯. FurthermorePP (B(H)) = PD(B(H)) = D.

Proof. To simplify notation let PP = PP (B(H)) and similarly for D. By The-orem 13 PP = PD, and by Theorem 12 PP = PP ♯ . Thus PD = PP ♯ . Note thatby Lemma 7

P ♯ = β ∈ P(H) : β α∗ ∈ CP (H), ∀α ∈ P.henceD ⊂ P ♯. Since PD = PP ♯ , a linear functional φ is positive on P (B(H), D)if and only if it is positive on P (B(H), P ♯). Since D ⊂ P ♯ it follows thatP (B(H), P ♯) ⊂ P (B(H), D), hence from the Hahn-Banach Theorem that they

are equal. Thus by Lemma 9 and Lemma 11 φ∗ ∈ P ♯ if and only if φ ≥ 0 onCP = P (B(H), P ♯) = P (B(H), D), if and only if φ ∈ PD = PP .

Let φ∗ ∈ P ♯, so that φ ∈ PD = PP , hence by Theorem 1 Cφ ∈ P (B(H), P ),which by Lemma 16 equals the cone E in Lemma 14. Thus Cφ = Cφ1 + Cφ2

with φ1 ∈ CP (H) and φ2 ∈ Cop(H), hence φ ∈ D, as is φ∗. Thus P ♯ ⊂ D,and they are equal. Since D is closed under the *-operation, so is P ♯, so bythe above D = P ♯ = PD. The proof is complete.

We conclude by showing the analogue of Lemma 8 for decomposable maps.When M = B(H) the result is a strengthening of the result in [5], which statesthat φ is decomposable if and only if ι⊗ φ is positive on F.

Corollary 19. Let ω denote the maximally entangled state on B(H), and letφ ∈ P(H). Then φ is decomposable if and only if ω (ι ⊗ φ) ≥ 0 on the coneF = x ∈ B(H ⊗H)+ : ι⊗ t(x) ≥ 0.Proof. By Theorem 18 and Lemma 17 φ ∈ D if and only if φ ∈ PD if and onlyif φ ≥ 0 on P (M,D) = F, if and only if

0 ≤ Tr(Cφx) = Tr(ι⊗ φ(p)x) = Tr(p(ι⊗ φ∗)(x)) = ω (ι⊗ φ∗)(x),for all x ∈ F. Since D is closed under *-operation, the corollary follows.

References

[1] I. Bengtsson and K. Zyczkowski, Geometry of quantum states, Cambridge Univ. Press,Cambridge, 2006. MR2230995 (2007k:81001)

[2] M. D. Choi, Completely positive linear maps on complex matrices, Linear Algebra andAppl. 10 (1975), 285–290. MR0376726 (51 #12901)

[3] M. Horodecki, P. Horodecki and R. Horodecki, Separability of mixed states: necessaryand sufficient conditions, Phys. Lett. A 223 (1996), no. 1-2, 1–8. MR1421501 (97k:81009)

[4] M. Horodecki, P. W. Shor and M. B. Ruskai, Entanglement breaking channels, Rev.Math. Phys. 15 (2003), no. 6, 629–641. MR2001114 (2005d:81053)

[5] E. Størmer, Decomposable positive maps on C∗-algebras, Proc. Amer. Math. Soc. 86

(1982), no. 3, 402–404. MR0671203 (84a:46123)



[6] E. Størmer, Extension of positive maps into B(H), J. Funct. Anal. 66 (1986), no. 2,235–254. MR0832990 (87f:46105)

[7] E. Størmer, Separable states and positive maps, J. Funct. Anal. 254 (2008), no. 8, 2303–2312. MR2402111 (2009c:46083)

[8] E. Størmer, Separable states and positive maps II. To appear in Math. Scand.[9] S.J.Szarek, E.Werner, and K.Zyczkowski, Geometry of sets of quantum maps: a

generic positive map acting on a high dimensional system is notcompletely positive.arixiv:0710.1571v2.

Received November 4, 2009; accepted April 1, 2009

Erling StørmerDepartment of Mathematics, University of Oslo, 0316 Oslo, Norway.E-mail: [email protected]




Completely positive maps of order zero

Wilhelm Winter and Joachim Zacharias



Abstract. We say a completely positive contractive map between two C∗-algebras has orderzero, if it sends orthogonal elements to orthogonal elements. We prove a structure theoremfor such maps. As a consequence, order zero maps are in one-to-one correspondence with∗-homomorphisms from the cone over the domain into the target algebra. Moreover, weconclude that tensor products of order zero maps are again order zero, that the compositionof an order zero map with a tracial functional is again a tracial functional, and that orderzero maps respect the Cuntz relation, hence induce ordered semigroup morphisms betweenCuntz semigroups.

1. Introduction

There are various types of interesting maps between C∗-algebras, all ofwhich can serve as morphisms of a category with objects (a subclass of) theclass of all C∗-algebras. As a first choice, continuous ∗-homomorphisms cometo mind, and it follows from spectral theory that in fact any ∗-homomorphismbetween C∗-algebras is automatically continuous, even contractive. At the op-posite end of the scale, one might simply consider (bounded) linear maps. It isthen natural to study classes of morphisms which lie between ∗-homomorphismsand linear maps. For example, one might ask a linear map to preserve theinvolution, or even the order structure, i.e., to be self-adjoint or positive, re-spectively. In noncommutative topology, it is also often desirable to considermaps which have well-behaved amplifications to matrix algebras; this leads tothe strictly smaller classes of completely bounded, or completely positive (c.p.)maps, for example. In contrast amplifications of ∗-homomorphisms automati-cally are ∗-homomorphisms.

Emphasizing the ∗-algebra structure rather than the order structure of aC∗-algebra, one might also consider Jordan ∗-homomorphisms (amplifications

Partially supported by EPSRC First Grant EP/G014019/1.

312 Wilhelm Winter and Joachim Zacharias

of which again are Jordan ∗-homomorphisms). Another concept, which hasrecently turned out to be highly useful, but has received less attention in theliterature, is that of orthogonality (or disjointness) preserving maps. By this,we mean linear maps which send orthogonal elements to orthogonal elements.There is a certain degree of freedom here since one might only ask for or-thogonality of supports or ranges to be preserved. This distinction becomesirrelevant in the case of c.p. maps. Instead of orthogonality preserving we willuse the term order zero.

In [16], Wolff proved a structure theorem for bounded, linear, self-adjoint,disjointness preserving maps with unital domains: Any such map is a com-pression of a Jordan ∗-homomorphism with a self-adjoint element commutingwith its image.

Later (but independently) the first named author arrived at a very simi-lar result for c.p. order zero maps in the case of finite-dimensional domains.Any such map is a compression of a ∗-homomorphism with a positive elementcommuting with its image. Order zero maps with finite-dimensional domainshave been used in [11], [14], [6] and [5] as building blocks of noncommutativepartitions of unity to define noncommutative versions of topological coveringdimension; see [10] and [12] for related applications. They will serve a similarpurpose in [15]. However, also order zero maps with more general domainsoccur in a natural way. To analyze these it will be crucial to have a structuretheorem for general c.p. order zero maps at hand.

In the present paper we use Wolff’s result to provide such a generalization,see Theorem 3.3. Compared to Wolff’s theorem, our result produces a strongerstatement from stronger hypotheses; it has the additional benefit that it coversthe nonunital situation as well.

We obtain a number of interesting consequences from Theorem 3.3. First, itturns out that completely positive contractive (c.p.c.) order zero maps from Ainto B are in one-to-one correspondence with ∗-homomorphisms from the coneover A into B. This point of view also leads to a notion of positive functionalcalculus for c.p. order zero maps. We then observe that tensor products of c.p.order zero maps are again order zero; this holds in particular for amplificationsof c.p. order zero maps to matrix algebras. Moreover, we show that thecomposition of a c.p. order zero map with a tracial functional again is atracial functional. Finally, we show that (unlike general c.p. maps) order zeromaps induce ordered semigroup morphisms between Cuntz semigroups. In fact,this observation is one of our motivations for studying order zero maps, since itshows that they provide a natural framework to study the question when mapsat the level of Cuntz semigroups can be lifted to maps betweenC∗-algebras. ForK-theory, this problem has been well-studied; it is of particular importance forthe classification program for nuclear C∗-algebras. While the Cuntz semigroupin recent years also has turned out to be highly relevant for the classificationprogram (cp. [4], [2]), at this point not even a bivariant version (resemblingKasparov’s KK -theory) has been developed. We are confident that our resultscan be used to build such a theory; this will be pursued in subsequent work.


Completely positive maps of order zero 313

Our paper is organized as follows. In Section 2 we recall some facts aboutorthogonality in C∗-algebras and introduce the notion of c.p. order zero maps.In Section 3, we prove a unitization result as well as our structure theorem forsuch maps. We derive a number of corollaries in Section 4.

We would like to thank the referee for a number of helpful comments.

2. Orthogonality

In this section we recall some facts about orthogonality in C∗-algebras,introduce the notion of c.p. order zero maps, and recall a result of Wolff.

Notation 2.1. Let a, b be elements in a C∗-algebra A. We say a and b areorthogonal, a ⊥ b, if ab = ba = a∗b = ab∗ = 0.

Remark 2.2. In the situation of the preceding definition, note that a ⊥ b ifand only if a∗a ⊥ b∗b, a∗a ⊥ bb∗, aa∗ ⊥ b∗b and aa∗ ⊥ bb∗.

Note also that, if a and b are self-adjoint, then a ⊥ b if and only if ab = 0.

Definition 2.3. Let A and B be C∗-algebras and let ϕ : A → B be a c.p.map. We say ϕ has order zero, if, for a, b ∈ A+,

a ⊥ b⇒ ϕ(a) ⊥ ϕ(b).

Remark 2.4. In the preceding definition, we could instead consider generalelements a, b ∈ A; this yields the same definition, since we assume ϕ to becompletely positive.

To see this, note that if ϕ respects orthogonality of arbitrary elements, ittrivially has order zero. Conversely, suppose ϕ has order zero, i.e., respectsorthogonality of positive elements, and let a ⊥ b ∈ A be arbitrary. Then,a∗a ⊥ b∗b, a∗a ⊥ bb∗, aa∗ ⊥ bb∗ and aa∗ ⊥ b∗b. We obtain ϕ(a∗a) ⊥ ϕ(b∗b),ϕ(a∗a) ⊥ ϕ(bb∗), ϕ(aa∗) ⊥ ϕ(bb∗) and ϕ(aa∗) ⊥ ϕ(b∗b). But since ϕ is c.p., wehave 0 ≤ ϕ(a∗)ϕ(a) ≤ ϕ(a∗a), 0 ≤ ϕ(a)ϕ(a∗) ≤ ϕ(aa∗) (and similarly for b inplace of a), which yields that ϕ(a∗)ϕ(a) ⊥ ϕ(b∗)ϕ(b), ϕ(a∗)ϕ(a) ⊥ ϕ(b)ϕ(b∗),ϕ(a)ϕ(a∗) ⊥ ϕ(b)ϕ(b∗) and ϕ(a)ϕ(a∗) ⊥ ϕ(b∗)ϕ(b), since orthogonality is ahereditary property. By Remark 2.2, this implies that ϕ(a) ⊥ ϕ(b), whence ϕrespects orthogonality of arbitrary elements.

Examples 2.5. Any ∗-homomorphism between C∗-algebras clearly has orderzero. More generally, if π : A → B is a ∗-homomorphism and h ∈ B is apositive element satisfying [h, π(A)] = 0, then ϕ( . ) := hπ( . ) defines a c.p.order zero map. We will show in Theorem 3.3 that any c.p. order zero map isessentially of this form.

2.6. In [16], Wolff defined a bounded linear map to be disjointness preserving,if it is self-adjoint and sends orthogonal self-adjoint elements to orthogonal self-adjoint elements. For the convenience of the reader, we state below the mainresult of that paper, [16, Theorem 2.3]. Recall that a Jordan ∗-homomorphismπ : A → B between C∗-algebras is a linear self-adjoint map preserving theJordan product a ·b = 1

2 (ab+ba). (Equivalently π preserves squares of positive

contractions, i.e., π(a2) = π(a)2 for all 0 ≤ a ≤ 1.)



Theorem. Let A and B be C∗-algebras, with A unital, and let ϕ : A→ B be adisjointness preserving map. Set C := ϕ(1A)ϕ(1A)′. Then, ϕ(A) ⊂ C andthere is a Jordan ∗-homomorphism π : A→M(C) from A into the multiplieralgebra of C satisfying

ϕ(a) = ϕ(1A)π(a)

for all a ∈ A.

3. The main result

Below, we prove a unitization result for c.p. order zero maps as well as ourmain theorem.

Notation 3.1. Following standard notation, we will write A+ for the 1-pointunitization of a C∗-algebra A, i.e., A+ = A⊕C as a vector space with the usualmultiplication rules. If ϕ : A→ B is a c.p.c. map into a unital C∗-algebra B,we write ϕ+ : A+ → B for the uniquely determined unital c.p. extension of ϕ.Recall that if ϕ is a ∗-homomorphism then so is ϕ+. We denote byM(A) themultiplier algebra of A and by A∗∗ its bidual, identified with the enveloppingvon Neumann algebra.

Proposition 3.2. Let A and B be C∗-algebras, with A nonunital, and letϕ : A→ B be a c.p.c. order zero map. Set C := C∗(ϕ(A)) ⊂ B.

Then, ϕ extends uniquely to a c.p.c. order zero map ϕ(+) : A+ → C∗∗.

Proof. We may clearly assume that C acts nondegenerately on a Hilbert spaceH.

Choose an increasing approximate unit (uλ)λ∈Λ for A and note that

(1) g := s.o. limλϕ(uλ) ∈ C∗∗

exists in the bidual of C since the ϕ(uλ) form a bounded, monotone increasingnet in C∗∗. Define a linear map

ϕ(+) : A+ → C∗∗

by

ϕ(+)(a) := ϕ(a), a ∈ Aand

(2) ϕ(+)(1A+) := g.

Note that ϕ(+) is well defined since A+ ∼= A⊕ C as a vector space.By Stinespring’s Theorem, there are a Hilbert space H1, a (nondegenerate)

∗-homomorphism

σ : A→ B(H1)

and an operator v ∈ B(H1,H) such that

v∗v ≤ 1H and ϕ(a) = v∗σ(a)v



for a ∈ A. Note that

g = s.o. limλϕ(uλ)

= s.o. limλv∗σ(uλ)v

= v∗(s.o.(limλσ(uλ)))v

= v∗1H1v

= v∗v,

where for the fourth equality we have used that σ is nondegenerate. Let

σ+ : A+ → B(H1)

be the unitization of σ, i.e.,

σ+(a+ α · 1A+) = σ(a) + α · 1H1 for a ∈ A, α ∈ C;

σ+ is a ∗-homomorphism. We have, for a ∈ A and α ∈ C,

v∗σ+(a+ α · 1A+)v = v∗σ(a)v + α · v∗v= ϕ(a) + α · g= ϕ(+)(a+ α · 1A+),

so ϕ(+) is c.p.c., being a compression of a ∗-homomorphism.We next check that ϕ(+) is again an order zero map. To this end, let

a+ α · 1A+ and b+ β · 1A+

in (A+)+ be orthogonal elements. Since orthogonality passes to quotients, wesee that at least one of α and β has to be zero. So let us assume β = 0 andnote that this implies b ≥ 0; note also that a = a∗ and α ≥ 0.

We have, for each λ ∈ Λ,

a+α·1A+ = (a+α·1A+)12 (1A+−uλ)(a+α·1A+)

12 +(a+α·1A+)

12uλ(a+α·1A+)

12 ,

with the second summand being an element of A dominated by a + α · 1A+ .This yields that

b ⊥ (a+ α · 1A+)12uλ(a+ α · 1A+)

12 ,

hence

(3) ϕ(+)(b)ϕ(+)((a+ α · 1A+)12 uλ(a+ α · 1A+)

12 ) = 0,

since ϕ(+) agrees with the order zero map ϕ on A.



Furthermore, using continuity of ϕ(+) and the fact that (uλ)Λ is approxi-mately central with respect to A+, we check that

0 ≤ s.o. limλϕ(+)((a+ α · 1A+)

12 (1A+ − uλ)(a+ α · 1A+)

12 )

= s.o. limλϕ(+)((1A+ − uλ)(a+ α · 1A+))

= s.o. limλϕ(+)(α · (1A+ − uλ))

= α · (ϕ(+)(1A+)− s.o. limλϕ(uλ))

(2),(1)= 0.(4)

We obtain

ϕ(+)(b)ϕ(+)(a+ α · 1A+)

= ϕ(+)(b)ϕ(+)((a+ α · 1A+)12uλ(a+ α · 1A+)

12 )

+ ϕ(+)(b)ϕ(+)((a+ α · 1A+)12 (1A+ − uλ)(a+ α · 1A+)

12 )

(3)= ϕ(+)(b)ϕ(+)((a+ α · 1A+)

12 (1A+ − uλ)(a+ α · 1A+)

12 )

s.o.→ 0

(where the last assertion follows from (4)), which implies that

ϕ(+)(b)ϕ(+)(a+ α · 1A+) = 0.

Therefore, ϕ(+) has order zero.To show that ϕ(+) is the unique c.p.c. order zero extension of ϕ (mapping

from A+ to C∗∗), suppose ψ : A+ → C∗∗ was another such extension, withd := ψ(1A+). Since ψ is positive, it is clear that

d = ψ(1A+) ≥ s.o. limλψ(uλ) = g.

Now, suppose that ‖d−g‖ > 0. Using that ϕ(uλ)1n → 1C∗∗ strongly as λ→∞

and n→∞, it is straightforward to show that there are η > 0 and λ ∈ Λ suchthat

‖(d− g)ϕ(uλ)(d− g)‖ ≥ η.Using functional calculus, one finds u,w ∈ A+ of norm at most one such that

‖u− uλ‖ < η/2

and

wu = u.

The latter implies that 1A+ −w and u are orthogonal elements in A+, whence

ψ(1A+ − w) ⊥ ψ(u) = ϕ(u) = ϕ(+)(u) ⊥ ϕ(+)(1A+ − w).



Combining these facts, we obtain

η ≤ ‖(d− g)ϕ(uλ)(d− g)‖≤ ‖(d− g)ϕ(uλ)‖≤ ‖(ψ(1A+)− ϕ(+)(1A+))ϕ(u)‖ +

η

2

= ‖(ψ(1A+ − (1A+ − w))− ϕ(+)(1A+ − (1A+ − w)))ϕ(u)‖ +η

2

= ‖(ψ(w) − ϕ(+)(w))ϕ(u)‖ +η

2

= ‖(ϕ(w) − ϕ(w))ϕ(u)‖ +η

2

=η

2,

a contradiction, so that d = g and ψ and ϕ(+) coincide.

Remark. It will follow from (the proof of) the next theorem that the rangeof the map ϕ(+) of the preceding proposition in fact lies inM(C).

3.3. Our main result is the following structure theorem for c.p. order zeromaps.

Theorem. Let A and B be C∗-algebras and ϕ : A→ B a c.p. order zero map.Set C := C∗(ϕ(A)) ⊂ B.

Then, there is a positive element h ∈ M(C) ∩ C′ with ‖h‖ = ‖ϕ‖ and a∗-homomorphism

πϕ : A→M(C) ∩ h′such that

πϕ(a)h = ϕ(a) for a ∈ A.If A is unital, then h = ϕ(1A) ∈ C.

Proof. By rescaling ϕ if necessary, we may clearly assume that ϕ is contractive.Let us first assume that A is unital, and set

(5) h := ϕ(1A) ∈ C.We may further assume that C acts nondegenerately on its universal Hilbertspace H.

By [16, Theorem 2.3(i)], we have h ∈ Z(C), since A is unital and ϕ is dis-jointness preserving. Moreover, one checks that h is a strictly positive elementof C, and since C ⊂ B(H) is nondegenerate, this implies that the supportprojection of h is 1H. On the other hand, the support projection of h can beexpressed as

(6) s.o. limn→∞

(h+1

n· 1H)−1h = 1H = 1C∗∗ .

We now define a mapπϕ : A→ C∗∗ ⊂ B(H)



by

(7) πϕ(a) := s.o. limn→∞

(h+1

n· 1H)−1ϕ(a).

Existence of the limit can be checked on positive elements, since then thesequence (h+ 1

n ·1H)−1ϕ(a) is monotone increasing. Since πϕ is a strong limitof c.p. maps, it is c.p. itself. Since h commutes with ϕ(A), one checks that πϕagain has order zero. Moreover,

(8) πϕ(1A) = 1H

by (6), so πϕ is unital.Now by [16, Lemma 3.3], πϕ is a Jordan ∗-homomorphism, so

(9) πϕ(a2) = πϕ(a)2 for a ∈ A+.

Moreover πϕ is u.c.p. and it is well-known that a completely positive Jor-dan homomorphism is a ∗-homomorphism (this follows for example from [1,Theorem II.6.9.18]). For the reader’s convenience we include the following ar-gument: by Stinespring’s Theorem applied to πϕ we may assume that there isa unital C∗-algebra D containing C∗∗ and a ∗-homomorphism

: A→ D

such that

(10) πϕ(a) = 1C∗∗(a)1C∗∗ for a ∈ A.

We now compute

‖1C∗∗(a)− 1C∗∗(a)1C∗∗‖2

= ‖1C∗∗(a)(1D − 1C∗∗)(a)1C∗∗‖(10)= ‖πϕ(a2)− πϕ(a)2‖

(9)= 0(11)

for a ∈ A+, whence

πϕ(ab)(10)= 1C∗∗(a)(b)1C∗∗

(11)= 1C∗∗(a)1C∗∗(b)1C∗∗

(10)= πϕ(a)πϕ(b)

for a, b ∈ A+. By linearity of πϕ it follows that πϕ is multiplicative, hence a∗-homomorphism.



Next, we check that for a ∈ A,

ϕ(a) − πϕ(a)h(7)= ϕ(a)− ϕ(a)s.o. lim

n→∞(h+

1

n· 1H)−1h

(5)= ϕ(a)− ϕ(a)s.o. lim

n→∞(h+

1

n· 1H)−1ϕ(1A)

(7)= ϕ(a)− ϕ(a)πϕ(1A)

(8)= 0,

so

ϕ(a) = πϕ(a)h = hπϕ(a)

for all a ∈ A, and πϕ(A) ⊂ h′.Finally, we have for a, b ∈ A

πϕ(a)ϕ(b) = πϕ(a)πϕ(b)h = πϕ(ab)h = ϕ(ab) ∈ C,and similarly ϕ(a)πϕ(b) = ϕ(ab), from which one easily deduces that

πϕ(A) ⊂M(C).

We have now verified the lemma in the case where A is unital. In the nonunitalcase, we may use Proposition 3.2 to extend ϕ to a c.p.c. order zero mapϕ(+) : A+ → C∗∗. By the first part of the proof there is a ∗-homomorphismπϕ(+) : A+ → C∗∗ such that ϕ(+)(a) = πϕ(+)(a)g = gπϕ(+)(a) for all a ∈ A+,

where g := ϕ(+)(1A+). Now if b ∈ A+, we have

gϕ(b) = gϕ(+)(b)

= gπϕ(+)(b)g

= gπϕ(+)(b12 )πϕ(+)(b

12 )g

= ϕ(+)(b12 )ϕ(+)(b

12 )

= ϕ(b12 )2 ∈ C,

which, by linearity, yields gϕ(b) ∈ C for any b ∈ A. From here it is straight-forward to conclude that g ∈ M(C), whence the images of ϕ(+) and πϕ(+) infact both live in M(C) by the first part of the proof. The ∗-homomorphismπϕ : A → M(C) will then just be the restriction of πϕ(+) : A+ → M(C)to A.

4. Some consequences

In this final section we derive some corollaries from Theorem 3.3.

Corollary 4.1. Let A and B be C∗-algebras, and ϕ : A → B a c.p.c. or-der zero map. Then, the map given by ϕ(id(0,1] ⊗ a) := ϕ(a) induces a ∗-homomorphism ϕ : C0((0, 1])⊗A→ B.

Conversely, any ∗-homomorphism : C0((0, 1]) ⊗ A → B induces a c.p.c.order zero map ϕ : A→ B via ϕ(a) := (id(0,1] ⊗ a).



These mutual assignments yield a canonical bijection between the spaces ofc.p.c. order zero maps from A to B and ∗-homomorphisms from C0((0, 1])⊗Ato B.

Proof. It is well known that C0((0, 1]) is canonically isomorphic to the universalC∗-algebra generated by a positive contraction, identifying id(0,1] with theuniversal generator.

Now if ϕ : A→ B is c.p.c. order zero, obtain C, h and πϕ from Theorem 3.3.There is a ∗-homomorphism

¯ : C0((0, 1])→M(C)

induced by

¯(id(0,1]) := h;

since h ∈ πϕ(A)′, ¯ and πϕ yield a ∗-homomorphism

ϕ : C0((0, 1])⊗A→M(C)

satisfying

ϕ(id(0,1] ⊗ a) = hπϕ(a) = ϕ(a) ∈ Cfor a ∈ A. Since C0((0, 1])⊗ A is generated by id(0,1] ⊗A as a C∗-algebra, wesee that in fact the image of ϕ lies in C ⊂ B.

Conversely, if

: C0((0, 1])⊗A→ B

is a ∗-homomorphism, then

ϕ( . ) := (id(0,1] ⊗ . )

clearly has order zero.That the assignments ϕ 7→ ϕ and 7→ ϕ are mutual inverses is straight-

forward to check.

4.2. As in [13], Theorem 3.3 allows us to define a positive functional calculusof c.p.c. order zero maps.

Corollary. Let ϕ : A → B be a c.p.c. order zero map, and let f ∈ C0((0, 1])be a positive function. Let C, h and πϕ be as in Theorem 3.3. Then, the map

f(ϕ) : A→ C ⊂ B,given by

f(ϕ)(a) := f(h)πϕ(a) for a ∈ A,is a well-defined c.p. order zero map. If f has norm at most one, then f(ϕ)is also contractive.

Proof. Since [h, πϕ(A)] = 0, we also have [f(h), πϕ(A)] = 0, which impliesthat f(ϕ) indeed is a c.p. map. Using that hπϕ(a) ∈ C for any a ∈ A, itis straightforward to conclude that f(h)πϕ(a) ∈ C for any a ∈ A. The laststatement is obvious.



Corollary 4.3. Let A, B, C and D be C∗-algebras and ϕ : A → B andψ : C → D c.p.c. order zero maps.

Then, the induced c.p.c. map

ϕ⊗µ ψ : A⊗µ C → B ⊗µ Dhas order zero, when ⊗µ denotes the minimal or the maximal tensor product.In particular, for any k ∈ N the amplification

ϕ(k) : Mk(A)→Mk(B)

has order zero.

Proof. Set

B := C∗(ϕ(A)) ⊂ B and D := C∗(ψ(C)) ⊂ D,and employ Theorem 3.3 to obtain ∗-homomorphisms

πϕ : A→M(B) and πψ : C →M(D)

and positive elements

hϕ ∈ M(B) and hψ ∈ M(D),

so that

ϕ(a) = πϕ(a)hϕ = hϕπϕ(a) and ψ(a) = πψ(c)hψ = hψπψ(c)

for a ∈ A and c ∈ C.Let us consider the maximal tensor product first. Fix a faithful nondegen-

erate representation

ι : B ⊗max D → B(H);

we have

ιmax : B ⊗max D → B ⊗ν D ⊂ B ⊗max D ⊂ B(H)

for some C∗-norm ν on B⊙D (ιmax is not necessarily injective). The represen-tation of B⊙ D on H yields representations of B and D on H with commutingimages (cp. [1, Theorem II.9.2.1]), and one observes that the induced repre-sentations of M(B) and M(D) on H also commute, and live in M(B ⊗ν D)(cp. [1, II.6.1.6]). We then obtain a ∗-homomorphism

ιmax :M(B)⊗maxM(D)→M(B ⊗ν D) ⊂ B(H)

extending ιmax, so that we may define a ∗-homomorphism

πmax : A⊗max Cπϕ⊗maxπψ−→ M(B)⊗maxM(D)

ιmax−→M(B ⊗ν D) ⊂ B(H)

and a positive element

hmax := ιmax (hϕ ⊗ hψ) ∈M(B ⊗ν D) ⊂ B(H).

It is straightforward to verify that

[hmax, πmax(A⊗max C)] = 0

and that

ϕ⊗max ψ : A⊗max C → B ⊗max D → B ⊗ν D ⊂ B ⊗max D



satisfies

ϕ⊗max ψ( . ) = πmax( . )hmax,

so ϕ⊗max ψ indeed has order zero.The minimal tensor product is handled similarly; only now we have to con-

sider faithful representations ιB : B → B(HB) and ιD : D → B(HD) whichinduce a (faithful) representation ιmin : M(B) ⊗minM(D) → B(HB ⊗ HD).As above, we set

πmin := ιmin (πϕ ⊗min πψ) : A⊗min C → B(HB ⊗HD)

and

hmin := ιmin (hϕ ⊗ hψ),

and check that

[hmin, πmin(A⊗min C)] = 0

and

ϕ⊗min ψ( . ) = πmin( . )hmin.

Corollary 4.4. Let A and B be C∗-algebras, ϕ : A → B a c.p.c. order zeromap, and τ a positive tracial functional on B.

Then, the composition τ ϕ is a positive tracial functional.The statement also holds when replacing the term ‘positive tracial functional’

with ‘2-quasitrace’ (in the sense of [8, 1.1.3]).

Proof. If τ is a positive tracial functional, we only need to check that τϕsatisfies the trace property. But, using the notation of 4.2, we have for a, b ∈ A,

τϕ(ab) = τ(ϕ12 (a)ϕ

12 (b))

= τ(ϕ12 (b)ϕ

12 (a))

= τϕ(ba).

Here, we have used the trace property of τ .If τ is only a 2-quasitrace, we also have to check two other things: First,

that τϕ extends to M2(A)—but this is obvious as ϕ extends to a c.p.c. orderzero map by Corollary 4.3. Second, we need to check that τϕ is additive oncommuting elements. However, if a, b ∈ A satisfy [a, b] = 0, then [ϕ(a), ϕ(b)] =h2πϕ([a, b]) = 0 and

τϕ(a+ b) = τ(ϕ(a) + ϕ(b)) = τϕ(a) + τϕ(b),

since ϕ is linear and τ is a 2-quasitrace.

4.5. The next result is one of our main motivations for studying order zeromaps in the abstract; it says that order zero maps induce maps at the level ofCuntz semigroups, since they respect the Cuntz relation; see [3], [7] and [9] foran introduction to Cuntz subequivalence and the Cuntz semigroup.



Corollary. Let A and B be C∗-algebras and ϕ : A → B a c.p.c. order zeromap.

Then, ϕ induces a morphism of ordered semigroups

W (ϕ) : W (A)→W (B)

between the Cuntz semigroups via

W (ϕ)(〈a〉) = 〈ϕ(k)(a)〉 if a ∈Mk(A)+.

Proof. Let a, b ∈ Mk(A)+ for some k ∈ N (it clearly suffices to consider thesame k for a and b) satisfying a - b. By definition of Cuntz subequivalence(cp. [3]), this means that there is a sequence (xn)N ⊂Mk(A) such that

a = limn→∞

x∗nbxn.

Letϕ(k) : Mk(A)→Mk(B)

denote the amplification of ϕ; note that ϕ(k) has order zero by Corollary 4.3.Let

C := C∗(ϕ(k)(Mk(A)))(∼= Mk(C∗(ϕ(A))))

and obtain from Theorem 3.3 a ∗-homomorphism

πϕ(k) : Mk(A)→M(C)

andh ∈ M(C)+

commuting with C. For n ∈ N, define

xn := h1n πϕ(k)(xn) = (ϕ(k))

1n (xn)

using 4.2; we then havexn ∈ C,

and

ϕ(k)(a) = limn→∞

ϕ(k)(x∗nbxn)

= limn→∞

πϕ(k)(x∗n)h12 πϕ(k)(b)h

12πϕ(k)(xn)

= limn→∞

x∗nϕ(k)(b)xn.

It follows that〈ϕ(k)(a)〉 - 〈ϕ(k)(b)〉,

so thatW (ϕ)(〈a〉) - W (ϕ)(〈b〉).

The argument also shows that if a ∼ b, then ϕ(k)(a) ∼ ϕ(k)(b), so that W (ϕ)indeed is well-defined and respects the order. Moreover, if a, b ∈ Mk(A) areorthogonal, then so are ϕ(k)(a), ϕ(k)(b) ∈Mk(B), whence

ϕ(k)(a⊕ b) = ϕ(k)(a)⊕ ϕ(k)(b)

and W (ϕ) is a semigroup morphism.



4.6. We remark in closing that if M and N are von Neumann algebras andϕ :M→N is a c.p.c order zero map, then the proof of 3.3 shows that ϕ = hπϕ,where h ∈ N+ commutes with the range of ϕ and πϕ is a ∗-homomorphismwhich is normal if ϕ is normal. Moreover, if ϕ : A→ B is any order zero c.p.cmap between C∗-algebras, then so is its bitransposed ϕ∗∗ : A∗∗ → B∗∗. Thisfollows for instance by bitransposing the factorization ϕ = (h1/2 · h1/2) πϕand usingM(C)∗∗ = C∗∗ ⊕ (M(C)/C)∗∗.

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Received March 17, 2009; accepted June 2, 2009

Wilhelm Winter and Joachim ZachariasSchool of Mathematical SciencesUniversity of Nottingham, Nottingham, NG7 2RD, UKE-mail: wilhelm.winter,[email protected]


Das Munster Journal of Mathematics ist elektronisch frei erhaltlich.Druckexemplare konnen uber das Mathematische Institut der UniversitatMunster bestellt werden. Gegenwartig betragt der Bezugspreis 100.- Europro Jahr.

The Munster Journal of Mathematics is electronically freely available.Printed issues can be ordered from the Mathematisches Institut der UniversitatMunster. Presently, the price is 100.- Euro per year.


Managing Editor:Linus KramerMath. Institut, Universitat MunsterEinsteinstr. 62, 48149 Munster, [email protected]

m¨unster journal of mathematics · bivariant k-theory have deepened our understanding of this...

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