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PACIFIC JOURNAL OF MATHEMATICS

Volume 222 No. 2 December 2005

PacificJournalofM

athematics

2005Vol.222,N

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PACIFIC JOURNAL OF MATHEMATICSVol. 222, No. 2, 2005

COMPLETELY POSITIVE INNER PRODUCTS AND STRONGMORITA EQUIVALENCE

HENRIQUE BURSZTYN AND STEFAN WALDMANN

We develop a general framework for the study of strong Morita equivalencein which C∗-algebras and hermitian star products on Poisson manifoldsare treated in equal footing. We compare strong and ring-theoretic Moritaequivalences in terms of their Picard groupoids for a certain class of unital∗-algebras encompassing both examples. Within this class, we show thatboth notions of Morita equivalence induce the same equivalence relation butgenerally define different Picard groups. For star products, this differenceis expressed geometrically in cohomological terms.

1. Introduction

This paper investigates several similarities between two types of algebras withinvolution: hermitian star products on Poisson manifolds and C∗-algebras. Theirconnection is suggested by their common role as “quantum algebras” in mathemat-ical physics, despite the fact that the former is a purely algebraic notion, whereasthe latter has important analytical features. Building on [Bursztyn and Waldmann2001a; 2001b], we develop in this paper a framework for their unified study, fo-cusing on Morita theory; in particular, the properties shared by C∗-algebras andstar products allow us to develop a general theory of strong Morita equivalence inwhich they are treated in equal footing.

Our set-up is as follows. We consider ∗-algebras over rings of the form C=R(i),where R is an ordered ring and i2 = −1. The main examples of R that we willhave in mind are R, with its natural ordering, and R[[λ]], with ordering induced by“asymptotic positivity”, i.e., a =

∑∞

r=0 arλr > 0 if and only if ar0 > 0, where ar0 is

the first nonzero coefficient of a. This general framework encompasses complex∗-algebras, such as C∗-algebras, as well as ∗-algebras over the ring of formal powerseries C[[λ]], such as hermitian star products. We remark that the case of ∗-algebrasover C has been extensively studied; see [Schmudgen 1990], for example, and[Waldmann 2004] for a comparison of notions of positivity.

MSC2000: 16D90, 46L08, 53D55.Keywords: complete positivity, strong Morita equivalence, Rieffel induction.Bursztyn thanks DAAD (German Academic Exchange Service) for financial support.

201

202 HENRIQUE BURSZTYN AND STEFAN WALDMANN

In our general framework, we define a purely algebraic notion of strong Moritaequivalence. The key ingredient in this definition is the notion of completely posi-tive inner products, which we use to refine Ara’s ∗-Morita equivalence [Ara 1999].One of our main results is that completely positive inner products behave well un-der the internal and external tensor products, and, as a consequence, strong Moritaequivalence defines an equivalence relation within the class of nondegenerate andidempotent ∗-algebras. This class of algebras includes both star products and C∗-algebras as examples. We prove that important constructions in the theory of C∗-algebras, such as Rieffel’s induction of representations [Rieffel 1974a], carry overto this purely algebraic setting, recovering and improving many of our previousresults [Bursztyn and Waldmann 2001a; 2001b].

In the ordinary setting of unital rings, Morita equivalence coincides with thenotion of isomorphism in the category whose objects are unital rings and mor-phisms are isomorphism classes of bimodules, composed via tensor product. Theinvertible arrows in this category form the Picard groupoid Pic [Benabou 1967],which is a “large” groupoid (in the sense that its collection of objects is not aset) encoding the essential aspects of Morita theory: the orbit of a ring in Pic

is its Morita equivalence class, whereas the isotropy groups in Pic are the usualPicard groups of rings. Analogously, we show that our purely algebraic notion ofstrong Morita equivalence coincides with the notion of isomorphism in a categorywhose objects are nondegenerate and idempotent ∗-algebras over a fixed ring C;morphisms and their compositions are given by more elaborate bimodules andtensor products, and the invertible arrows in this category form the strong Picardgroupoid Picstr. When restricted to C∗-algebras, we show that Picstr defines anequivalence relation which turns out to coincide with Rieffel’s (analytical) notionof strong Morita equivalence [Rieffel 1974b], and its isotropy groups are the Picardgroups of C∗-algebras as in [Brown et al. 1977]; these results are proven along thelines of [Ara 2001; Bursztyn and Waldmann 2001b].

In the last part of the paper, we compare strong and ring-theoretic Morita equiv-alences for unital ∗-algebras over C by analyzing the canonical groupoid morphism

(1–1) Picstr−→ Pic.

We prove that, for a suitable class of unital ∗-algebras, including both unital C∗-algebras and hermitian star products, Picstr and Pic have the same orbits, i.e., thetwo notions of Morita equivalence define the same equivalence relation. This is asimultaneous extension of Beer’s result [Beer 1982], in the context of C∗-algebras,and [Bursztyn and Waldmann 2002, Thm. 2], for deformation quantization. De-spite the coincidence of orbits, we show that, for both unital C∗-algebras and her-mitian star products, the isotropy groups of Pic and Picstr are generally different.We note that the obstructions to (1–1) being an equivalence can be described in

COMPLETELY POSITIVE INNER PRODUCTS. STRONG MORITA EQUIVALENCE 203

a unified way for both classes of ∗-algebras, due to common properties of theirautomorphism groups. A key ingredient for this discussion in the context of formaldeformation quantization is the fact that hermitian star products are always (com-pletely) positive deformations, in the sense that positive measures on the manifoldcan be deformed into positive linear functionals of the star product; see [Bursztynand Waldmann 2000b; 2004b].

The paper is organized as follows: In Section 2 we recall the basic definitionsand properties of ∗-algebras over ordered rings and pre-Hilbert spaces. Section3 is devoted to completely positive inner products, a central notion throughoutthe paper. In Section 4 we define various categories of representations and provethat internal and external tensor products of completely positive inner products areagain completely positive. In Section 5 we define strong Morita equivalence, provethat it is an equivalence relation within the class of nondegenerate and idempotent∗-algebras and show that strong Morita equivalence implies the equivalence ofthe categories of representations introduced in Section 4. In Section 6 we definethe strong Picard groupoid and relate our algebraic definition to the C∗-algebraicPicard groupoid, proving their equivalence. In Section 7 we study the map (1–1)for a suitable class of unital ∗-algebras. Finally, in Section 8, we consider hermitiandeformations, and, in particular, hermitian star products.

Conventions. Throughout this paper C will denote a ring of the form R(i), whereR is an ordered ring and i2 = −1. Unless otherwise stated, algebras and moduleswill always be over a fixed ring C. For a manifold M , C∞(M) denotes its algebraof complex-valued smooth functions.

2. ∗-Algebras, positivity and pre-Hilbert spaces

A ∗-algebra over C is a C-algebra equipped with an anti-linear involutive anti-automorphism. If A is a ∗-algebra over C, then there are natural notions of positivityinduced by the ordering structure on R: A positive linear functional is a C-linearmap ω : A → C satisfying ω(a∗a)≥ 0 for all a ∈ A, and an algebra element a ∈ A

is called positive if ω(a) ≥ 0 for all positive linear functionals ω of A. Elementsof the form

(2–1) r1a∗

1a1 + · · · + rna∗

nan,

ri ∈ R+, ai ∈ A are clearly positive. The set of positive algebra elements is denotedby A+; see [Bursztyn and Waldmann 2001a, Sec. 2] for details. These definitionsrecover the standard notions of positivity when A is a C∗-algebra; for A=C∞(M),positive linear functionals coincide with positive Borel measures on M with com-pact support, and positive elements are positive functions [Bursztyn and Waldmann2001a, App. B].


A linear map φ : A → B, where A and B are ∗-algebras over C, is called positiveif φ(A+)⊆ B+, and completely positive if the canonical extensions φ : Mn(A)→

Mn(B) are positive for all n ∈ N.

Example 2.1. Consider the maps tr : Mn(A)→ A and τ : Mn(A)→ A defined by

(2–2) tr(A)=

n∑i=1

Ai i , and τ(A)=

n∑i, j=1

Ai j ,

where A = (Ai j )∈ Mn(A). A direct computation shows that both maps are positive.Replacing A by MN (A) and using the identification Mn(MN (A)) ∼= MNn(A), itimmediately follows that tr and τ are completely positive maps.

A pre-Hilbert space H over C is a C-module with a C-valued sesquilinear innerproduct satisfying

(2–3) 〈φ,ψ〉 = 〈ψ, φ〉 and 〈φ, φ〉> 0 for φ 6= 0;

see [Bursztyn and Waldmann 2001a]. We use the convention that 〈 · , · 〉 is linear inthe second argument. These are direct analogues of complex pre-Hilbert spaces. A∗-representation of a ∗-algebra A on a pre-Hilbert space H is a ∗-homomorphismfrom A into the adjointable endomorphisms B(H) of H [Bursztyn and Waldmann2001a; 2001b]; the main examples are the usual representations of C∗-algebras onHilbert spaces and the formal representations of star products. See, for instance,[Bordemann and Waldmann 1998; Waldmann 2002].

3. Completely positive inner products

3A. Inner products and complete positivity. Let A be a ∗-algebra over C and con-sider a right A-module E. Throughout this paper, A-modules are always assumedto have a compatible C-module structure.

Remark 3.1. When A is unital, we adopt the convention that x · 1 = x for x ∈ E;morphisms between unital algebras are assumed to be unital.

An A-valued inner product on E is a C-sesquilinear (linear in the second argu-ment) map 〈 · , · 〉 : E × E → A so that, for all x, y ∈ E and a ∈ A,

(3–1) 〈x, y〉 = 〈y, x〉∗ and 〈x, y · a〉 = 〈x, y〉a.

The definition of an A-valued inner product on a left A-module is analogous, butwe require linearity in the first argument. We call an inner product 〈 · , · 〉 nonde-generate if 〈x, y〉 = 0 for all y implies that x = 0, and strongly nondegenerate ifthe map E → HomA(E,A), x 7→ 〈x, · 〉 is a bijection. Two inner products 〈 · , · 〉1

and 〈 · , · 〉2 on E are called isometric if there exists a module automorphism U with〈U x,U y〉1 = 〈x, y〉2.


An endomorphism T ∈ EndA(E) is adjointable with respect to 〈 · , · 〉 if thereexists T ∗

∈ EndA(E) (called an adjoint of T ) such that

(3–2) 〈x, T y〉 = 〈T ∗x, y〉

for all x, y ∈ E. The algebra of adjointable endomorphisms is denoted by BA(E),or simply B(E). If 〈 · , · 〉 is nondegenerate, then adjoints are unique and BA(E)

becomes a ∗-algebra over C. One defines the C-module BA(E,F) of adjointablehomomorphisms E → F analogously.

An inner product 〈 · , · 〉 on E is positive if 〈x, x〉 ∈ A+ for all x ∈ E, and positivedefinite if 0 6= 〈x, x〉 ∈ A+ for x 6= 0.

Definition 3.2. Consider En as a right Mn(A)-module, and let 〈 · , · 〉(n) be theMn(A)-valued inner product on En defined by

(3–3) 〈x, y〉(n)i j = 〈xi , y j 〉,

where x = (x1, . . . , xn) and y = (y1, . . . , yn)∈ En . We say that 〈 · , · 〉 is completelypositive if 〈 · , · 〉(n) is positive for all n.

Remark 3.3. Although the direct sum of nondegenerate (resp. positive, completelypositive) inner products is nondegenerate (resp. positive, completely positive), thismay not hold for positive definiteness: Consider A = Z2 as ∗-algebra over Z(i)[Bursztyn and Waldmann 2001b, Sec. 2]; then the canonical inner product on A ispositive definite but on A2 the vector (1, 1) satisfies 〈(1, 1), (1, 1)〉 = 1+ 1 = 0.

The following observation provides a way to detect algebras A for which posi-tive A-valued inner products on arbitrary A-modules are automatically completelypositive.

Proposition 3.4. Let A be a ∗-algebra satisfying the following property: for anyn ∈ N, if (Ai j ) ∈ Mn(A) satisfies

∑i j a∗

i Ai j a j ∈ A+ for all (a1, . . . , an) ∈ An ,then A ∈ Mn(A)

+. Then any positive A-valued inner product on an A-module isautomatically completely positive.

Proof. Let E be an A-module with positive inner product 〈 · , · 〉, and let x1, . . . , xn ∈

E. For a1, . . . , an ∈ A, the matrix A = (〈xi , x j 〉) satisfies∑i j

a∗

i 〈xi , x j 〉a j =

∑i j〈xi · ai , x j · a j 〉 =

⟨∑i

xi · ai ,∑

jx j · a j

⟩∈ A+.

So the matrix (〈xi , x j 〉) is positive and 〈 · , · 〉 is completely positive. �

The converse also holds, e.g., if A is unital.Note that, although a positive definite inner product is always nondegenerate, a

positive inner product which is nondegenerate may fail to be positive definite. This


is due to the fact that the degeneracy space of an A-module E with inner product〈 · , · 〉, defined by

(3–4) E⊥= {x ∈ E | 〈x, ·〉 = 0},

might be strictly contained in the space

(3–5) {x ∈ E | 〈x, x〉 = 0}.

Example 3.5. Let A=∧

•(Cn) be the Grassmann algebra over Cn , with ∗-involution

defined by e∗

i = ei , where e1, . . . , en is the canonical basis for Cn . Regard A asa right module over itself, equipped with inner product 〈x, y〉 = x∗

∧ y. Then〈ei , ei 〉 = 0. However, A⊥

= {0}, since 〈1, x〉 = x .

Any A-valued inner product on E induces a nondegenerate one on the quotientE/

E⊥. Moreover, (completely) positive inner products induce (completely) posi-tive inner products. In case E⊥

= {x ∈ E | 〈x, x〉 = 0}, the quotient inner productis positive definite.

∗-Algebras possessing a “large” amount of positive linear functionals, such asC∗-algebras and formal hermitian deformation quantizations [Bursztyn and Wald-mann 2001a; 2001b], are such that (3–4) and (3–5) coincide.

Example 3.6. Let A be a ∗-algebra over C with the property that, for any nonzerohermitian element a ∈ A, there exists a positive linear functional ω with ω(a) 6= 0.Under the additional assumption that 2 ∈ R is invertible, any A-module E withA-valued inner product is such that (3–4) and (3–5) coincide. The proof followsfrom the arguments in [Bursztyn and Waldmann 2001a, Sect. 5].

3B. Examples of completely positive inner products. Inner products on complexpre-Hilbert spaces are always completely positive. This result extends in two di-rections: on one hand, one can replace C by arbitrary rings C; on the other hand,C can be replaced by more general C∗-algebras.

Example 3.7 (Pre-Hilbert spaces over C). If A = C, then [Bursztyn and Waldmann2001a, Prop. A.4] shows that the condition in Proposition 3.4 is satisfied. So apositive C-valued inner product on any C-module H is completely positive. This isthe case, in particular, for inner products on pre-Hilbert spaces over C (which arenondegenerate).

Example 3.8 (Pre-Hilbert C∗-modules). Let A be a C∗-algebra over C = C.Then the condition in Proposition 3.4 holds; see [Raeburn and Williams 1998,Lem. 2.28]. So a positive A-valued inner product 〈 · , · 〉 on any A-module E iscompletely positive (see also [Raeburn and Williams 1998, Lem. 2.65]). When〈 · , · 〉 is positive definite, (E, 〈 · , · 〉) is called a pre-Hilbert C∗-module over A.


Example 3.7 uses the quotients fields of R and C, whereas Example 3.8 uses thefunctional calculus of C∗-algebras, so neither immediately extend to inner productswith values in arbitrary ∗-algebras. Nevertheless, one can still show the completepositivity of particular inner products.

Example 3.9 (Free modules). Consider AN as a right A-module with respect toright multiplication, equipped with the canonical inner product

(3–6) 〈x, y〉 =

N∑i=1

x∗

i yi ,

where x = (x1, . . . , xN ), y = (y1, . . . , yN )∈ AN . This inner product is completelypositive since, for x (1), . . . , x (n) ∈ AN , the matrix X =

(〈x (α), x (β)〉

)∈ Mn(A) can

be written as

(3–7) X =

∑i

X∗

i X i , where X i =

x (1)i · · · x (n)i

0 · · · 0...

...

0 · · · 0

.Note, however, that the inner product (3–6) need not be positive definite as theremay exist elements a ∈ A with a∗a = 0. If A is unital then (3–6) is stronglynondegenerate; in the nonunital case it may be degenerate.

Remark 3.10. Let E be an A-module with inner product 〈 · , · 〉 which can bewritten as

(3–8) 〈x, y〉 =

m∑i=1

Pi (x)∗ Pi (y), for x, y ∈ E,

where Pi : E → A are A-linear maps. By replacing x (α)i with Pi (x (α)) in Example3.9, one immediately sees that (3–8) is completely positive.

A direct computation shows that completely positive inner products restrict tocompletely positive inner products on submodules.

Example 3.11 (Hermitian projective modules). The restriction of the canonicalinner product (3–6) to any submodule of An is completely positive. In particular,hermitian projective modules, i.e., modules of the form E = PAn , where P ∈

Mn(A), P = P2= P∗, have an induced completely positive inner product (this

also follows from Remark 3.10). If A is unital, then this inner product is stronglynondegenerate.

The following simple observation concerns uniqueness.


Lemma 3.12. Let E be an A-module equipped with a strongly nondegenerate A-valued inner product 〈 · , · 〉. Let 〈 · , · 〉′ be another inner product on E. Then thereexists a unique hermitian element H ∈ B(E) such that

(3–9) 〈x, y〉′= 〈x, H y〉,

and 〈 · , · 〉′ is isometric to 〈 · , · 〉 if there exists an invertible U ∈ B(E) with H =

U∗U .

Example 3.13 (Hermitian vector bundles). Let A = C∞(M) be the algebra ofsmooth complex-valued functions on a manifold M . As a result of the Serre–Swantheorem [Swan 1962], hermitian projective modules PAN correspond to (sectionsof) vector bundles over M (since in this case idempotents are always equivalent toprojections — see Section 7A), and A-valued inner products correspond to hermit-ian fiber metrics.

As noticed in Example 3.11, there is a strongly nondegenerate inner product〈 · , · 〉 on PAN . For any other inner product 〈 · , · 〉′, there exists a unique hermitianelement H ∈ P MN (A)P such that 〈x, y〉

′= 〈x, H y〉. Since any positive invertible

element H ∈ P MN (C∞(M))P can be written as H = U∗U for an invertible U ∈

P MN (C∞(M))P , it follows from Lemma 3.12 that there is only one fiber metricon a vector bundle over M up to isometric isomorphism. We will generalize thisexample in Section 7A.

Example 3.14 (Nontrivial inner products). Even if the algebra A is a field, onecan have nontrivial inner products. For example, consider R = Q and C = Q(i).Then 3 ∈ C is a positive invertible element but there is no z ∈ C with zz = 3(write z = a + ib with a = r/n, b = s/n with r, s, n ∈ N, then take the equation3n2

= r2+s2 modulo 4). Hence 〈z, w〉

′= 3zw is completely positive and strongly

nondegenerate but not isometric to the canonical inner product 〈z, w〉 = zw.

4. Representations and tensor products

4A. Categories of ∗-representations. We now discuss the algebraic analogues ofHilbert C∗-modules; see [Lance 1995], for example. Let D be a ∗-algebra over C.

Definition 4.1. A (right) inner-product D-module is a pair (H, 〈 · , · 〉), where H isa (right) D-module and 〈 · , · 〉 is a nondegenerate D-valued inner product. If 〈 · , · 〉

is completely positive, we call (H, 〈 · , · 〉) a pre-Hilbert D-module.

Whenever there is no risk of confusion, we will denote an inner-product module(or pre-Hilbert module) simply by H.

We now consider ∗-representations of ∗-algebras on inner-product modules, ex-tending the discussion in [Bursztyn and Waldmann 2001a; Bursztyn and Waldmann


2001b; Bordemann and Waldmann 1998]. Let A be a ∗-algebra over C, and let H

be an inner-product D-module.

Definition 4.2. A ∗-representation of A on H is a ∗-homomorphism π : A →

BD(H).

An intertwiner between two ∗-representations (H, π) and (K, %) is an isometryT ∈ BD(H,K) such that, for all a ∈ A,

(4–1) Tπ(a)= %(a)T .

We denote by ∗-modD(A) the category whose objects are ∗-representations ofA on inner-product modules over D and morphisms are intertwiners. The sub-category whose objects are ∗-representations on pre-Hilbert modules is denotedby ∗-repD(A). Since both categories contain trivial representations of A, we willconsider the following further refinement: A ∗-representation (H, π) is stronglynondegenerate if

(4–2) π(A)H = H,

(by Remark 3.1, this is always the case if A is unital). The category of strongly non-degenerate ∗-representations of A on inner-product (resp. pre-Hilbert) D-modulesis denoted by ∗-ModD(A) (resp. ∗-RepD(A)).

Definition 4.3. An inner-product D-module H together with a strongly nondegen-erate ∗-representation of A will be called an (A,D)-inner-product bimodule; it isan (A,D)-pre-Hilbert bimodule if H is a pre-Hilbert module.

These are algebraic analogues of Hilbert bimodules as e.g. in [Landsman 2001,Def. 3.2]. This terminology differs from the one in [Ara 2001].

An isomorphism of inner-product (bi)modules (or pre-Hilbert (bi)modules) isjust a (bi)module homomorphism preserving inner products.

More generally, suppose AHD is a bimodule equipped with an arbitrary D-valuedinner product 〈 · , · 〉. We say that 〈 · , · 〉 is compatible with the A-action if

(4–3) 〈a · x, y〉 = 〈x, a∗· y〉,

for all a ∈ A and x, y ∈ H. Clearly, any ∗-representation of A on an inner-productmodule H over D makes H into a bimodule for which 〈 · , · 〉 and the A-action arecompatible. Unless otherwise stated, inner products on bimodules are assumed tobe compatible with the actions.

4B. Tensor products and Rieffel induction of representations. Let A and B be∗-algebras over C. Let FB be a right B-module equipped with a B-valued innerproduct 〈 · , · 〉F

B, and let BEA be a bimodule equipped with an A-valued inner prod-uct 〈 · , · 〉E

A compatible with the B-action. Following Rieffel [1974a; 1974b], there


is a well-defined A-valued inner product⟨· , ·

⟩F⊗E

Aon the tensor product FB⊗B BEA

completely determined by

(4–4)⟨y1 ⊗B x1, y2 ⊗B x2

⟩F⊗E

A=⟨x1, 〈y1, y2〉

FB · x2

⟩EA

for x1, x2 ∈ BEA and y1, y2 ∈ FB (we extend it to arbitrary elements using C-sesquilinearity). An analogous construction works for left modules carrying innerproducts.

If (FB ⊗B BEA)⊥ is the degeneracy space associated with

⟨· , ·

⟩F⊗E

A, then the

quotient(FB ⊗B BEA)

/(FB ⊗B BEA)

⊥

acquires an induced inner product, also denoted by⟨· , ·

⟩F⊗E

A, which is nondegen-

erate by Section 3A. Thus the pair

(4–5) FB ⊗B BEA :=

((FB ⊗B BEA)

/(FB ⊗B BEA)

⊥,⟨· , ·

⟩F⊗E

A

)is an inner-product A-module called the internal tensor product of (FB, 〈 · , · 〉F

B)

and ( BEA, 〈 · , · 〉EA). As we will see, in many examples the degeneracy space of

(4–4) is already trivial.

Lemma 4.4. If C is a ∗-algebra and CFB is a bimodule so that 〈 · , · 〉FB is compatible

with the C-action, then FB ⊗B BEA carries a canonical left C-action, compatiblewith

⟨· , ·

⟩F⊗E

A.

The proof of this lemma is a direct computation. It is also simple to check thatinternal tensor products have associativity properties similar to those of ordinary(algebraic) tensor products: Let GC be a C-module with C-valued inner product,and let CFB (resp. BEA) be a bimodule with B-valued (resp. A-valued) inner prod-uct compatible with the C-action (resp. B-action).

Lemma 4.5. There is a natural isomorphism

(4–6) ( GC ⊗C CFB) ⊗B BEA∼= GC ⊗C ( CFB ⊗B BEA)

induced from the usual associativity of algebraic tensor products.

Internal tensor products also behave well with respect to maps.

Lemma 4.6. Let CFB, CF′B be equipped with compatible B-valued inner products,

and let BEA, BE′A be equipped with compatible A-valued inner products. Let

S ∈ B( CFB, CF′B) and T ∈ B( BEA, BE′

A) be adjointable bimodule morphisms.Then their algebraic tensor product S ⊗B T induces a well-defined adjointablebimodule morphism S ⊗B T : CFB ⊗B BEA → CF′

B ⊗B BE′A with adjoint given by

S∗⊗B T ∗. If S and T are isometric then S ⊗B T is isometric as well.


Hence, for a fixed triple of ∗-algebras A, B and C, we obtain a functor

(4–7) ⊗B :∗-modB(C)×

∗-modA(B)−→∗-modA(C).

In the case of unital algebras, one can replace ∗-mod by ∗-Mod in (4–7).A central question is whether one can restrict the functor ⊗B to representations

on pre-Hilbert modules, or whether the tensor product (4–4) of two positive innerproducts remains positive. This is the case, for example, in the realm of C∗-algebras, but the proof uses the functional calculus; see [Raeburn and Williams1998, Prop. 2.64], for instance. Fortunately, a purely algebraic result can be ob-tained if one requires the inner products to be completely positive.

Theorem 4.7. If the inner products 〈 · , · 〉FB on FB and 〈 · , · 〉E

A on BEA are com-pletely positive, then the inner product

⟨· , ·

⟩F⊗E

Aon FB ⊗B BEA defined by (4–4) is

also completely positive.

Proof. Let 8(1), . . . 8(n) ∈ FB ⊗B BEA. We must show that the matrix

A =(⟨8(α),8(β)

⟩F⊗E

A

)is a positive element in Mn(A). Without loss of generality, we can write 8(α) =∑N

i=1 y(α)i ⊗B x (α)i , where N is the same for all α. Consider the map

(4–8) f : Mn(MN (B))−→ Mn(MN (A)), (Bαβi j ) 7→(⟨

x (α)i , Bαβi j · x (β)j

⟩EA

),

1 ≤ i, j ≤ N , 1 ≤ α, β ≤ n. We claim that f is a positive map. Indeed, as aconsequence of the definition of positive maps in Section 2, it suffices to show thatf (B∗B) is positive for any B ∈ Mn(MN (B)). A direct computation shows that,for B = (Bαβi j ),

f (B∗B)=

N∑k=1

n∑γ=1

Cγ

k with(Cγ

k

)αβi j =

⟨Bγαki x (α)i , Bγβk j x (β)j

⟩EA,

which is positive since 〈 · , · 〉EA is completely positive. Since the matrix(⟨y(α)i , y(β)j

⟩FB

)∈ MnN (B)

is positive, for 〈 · , · 〉FB in F is completely positive, it follows that the matrix(⟨

x (α)i ,⟨y(α)i , y(β)j

⟩FB· x (β)j

⟩EA

)is a positive element in MnN (A). Since summation over i, j defines a positive mapτ : MnN (A)→ Mn(A), see Example 2.1, the matrix

(4–9)N∑

i, j=1

(⟨x (α)i ,

⟨y(α)i , y(β)j

⟩FB· x (β)j

⟩EA

)=(⟨8(α),8(β)

⟩F⊗E

A

)= A


is positive. This concludes the proof. �

As pointed out in Section 3A, if 〈 · , · 〉F⊗EA is completely positive, so is the in-

duced inner product on FB ⊗B BEA.

Corollary 4.8. If FB and BEA have completely positive inner products, thenFB ⊗B BEA is a pre-Hilbert module.

It follows that the functor ⊗B in (4–7) restricts to a functor

(4–10) ⊗B :∗-repB(C)×

∗-repA(B)−→∗-repA(C),

and, from (4–10), we obtain two functors by fixing each one of the two arguments.

Example 4.9 (Rieffel induction). Let A, B and D be ∗-algebras, and fix a (B,A)-bimodule BEA ∈

∗-repA(B). We then have a functor

(4–11) RE = BEA ⊗A · :∗-repD(A)−→

∗-repD(B);

on objects, RE( AHD) = BEA ⊗A AHD, and, on morphisms, RE(T ) = id ⊗AT , forT ∈ B(H,H′).

This functor is called Rieffel induction and relates the representation theories ofA and B on pre-Hilbert modules over a fixed ∗-algebra D.

Example 4.10 (Change of base ring). Similarly, we can change the base algebra D

in ∗-repD(A): Let A, D and D′ be ∗-algebras and let DGD′ ∈∗-repD′(D). Then ⊗D

induces a functor

(4–12) SG = · ⊗D DGD′ :∗-repD(A)−→

∗-repD′(A)

defined analogously to (4–11).

A direct consequence of Lemma 4.5 is that the following diagram commutes upto natural transformations:

(4–13)

∗-repD(A)SG- ∗-repD′(A)

∗-repD(B)

RE

?SG- ∗-repD′(B)

RE

?

Remark 4.11 (Rieffel induction for C∗-algebras). In the original setting of C∗-algebras, Rieffel’s construction [1974a; 1974b] relates categories of ∗-representa-tions on Hilbert spaces (in particular, D = C = C), so one needs to consider anextra completion of BEA ⊗A AHD with respect to the norm induced by (4–4). Since∗-representations of C∗-algebras on pre-Hilbert spaces are necessarily bounded,


this completion is canonical, so one recovers Rieffel’s original construction fromthis algebraic approach; see [Bursztyn and Waldmann 2001b]. More generally, inthis setting, D could be an arbitrary C∗-algebra.

Examples of algebraic Rieffel induction of ∗-representations in the setting offormal deformation quantization can be found in [Bursztyn and Waldmann 2000a;2002], for instance.

Remark 4.12 (External tensor products). Let Ai and Bi be ∗-algebras over C,i = 1, 2. The tensor products A = A1 ⊗C A2 and B = B1 ⊗C B2 are naturally∗-algebras. Let Ei be (Bi ,Ai )-bimodules for i = 1, 2 and consider the (B,A)-bimodule E = E1 ⊗C E2. If each Ei is endowed with an Ai -valued inner product〈 · , · 〉i , compatible with the Bi -action, then we have an inner product 〈 · , · 〉 on E,compatible with the B-action, uniquely defined by

(4–14) 〈x1 ⊗C x2, y1 ⊗C y2〉 = 〈x1, y1〉1 ⊗C 〈x2, y2〉2

for xi , yi ∈ Ei . We call the inner product defined by (4–14) the external tensorproduct of 〈 · , · 〉1 and 〈 · , · 〉2. Just as for internal tensor products, if 〈 · , · 〉i arecompletely positive, then so is 〈 · , · 〉. The construction is also functorial in a senseanalogous to Lemma 4.6.

5. Strong Morita equivalence

5A. Definition. An A-valued inner product 〈 · , · 〉EA on an A-module E is called

full if

(5–1) C-span{〈x, y〉EA | x, y ∈ E} = A.

Let A and B be ∗-algebras over C.

Definition 5.1. Let BEA be a (B,A)-bimodule with an A-valued inner product〈 · , · 〉E

A and a B-valued inner product B〈 · , · 〉E. We call(

BEA, B〈 · , · 〉E, 〈 · , · 〉EA

)a

∗-equivalence bimodule if

(1) 〈 · , · 〉EA (resp. B〈 · , · 〉E) is nondegenerate, full and compatible with the B-

action (resp. A-action);

(2) For all x, y, z ∈ E one has x · 〈y, z〉EA = B〈x, y〉

E· z;

(3) B · E = E and E · A = E.

If 〈 · , · 〉EA and B〈 · , · 〉E are completely positive, then BEA is called a strong equiv-

alence bimodule.

Whenever the context is clear, we will refer to strong or ∗-equivalence bimodulessimply as equivalence bimodules.


Definition 5.2. Two ∗-algebras A and B are ∗-Morita equivalent (resp. stronglyMorita equivalent) if there exists a ∗- (resp. strong) (B,A)-equivalence bimodule.

The definition of ∗-Morita equivalence goes back to [Ara 1999]. Since thisnotion does not involve positivity, its definition makes sense for ground rings notnecessarily of the form C = R(i).

Remark 5.3 (Formal Morita equivalence of ∗-algebras). In [Bursztyn and Wald-mann 2001a] we had a more technical formulation of strong Morita equivalencefor ∗-algebras over C, called formal Morita equivalence. Definition 5.2, based oncompletely positive inner products, is conceptually clearer (though, at least forunital algebras, it is equivalent to the one in [Bursztyn and Waldmann 2001a]) andyields refinements of the results in that paper.

Remark 5.4 (Strong Morita equivalence of C∗-algebras). The definition given in[Rieffel 1974b] for a strong equivalence bimodule of C∗-algebras (see also [Rae-burn and Williams 1998]) is a refinement of Definition 5.1 involving topologicalcompletions which do not make sense in a purely algebraic setting. Nevertheless,one recovers Rieffel’s notion as follows [Ara 2001], [Bursztyn and Waldmann2001b, Lem. 3.1]: Two C∗-algebras are strongly Morita equivalent in Rieffel’ssense if and only if their minimal dense ideals are strongly Morita equivalent (or∗-Morita equivalent) in the sense of Definition 5.1. In particular, for minimal denseideals of C∗-algebras, ∗- and strong Morita equivalences coincide (see Section 6B).

As we now discuss, ∗- and strong Morita equivalences are in fact equivalencerelations for a large class of ∗-algebras.

Lemma 5.5. The notions of ∗- and strong Morita equivalences define a symmetricrelation.

For the proof, we just note that if BEA is a ∗- (resp. strong) (B,A)-equivalence bi-module, then its conjugate bimodule AEB is an ∗- (resp. strong) (A,B)-equivalencebimodule. See [Bursztyn and Waldmann 2001a, Sect. 5].

For reflexivity and transitivity, one needs to be more restrictive. Recall that analgebra A is nondegenerate if a ∈ A, A ·a = 0 or a ·A = 0 implies that a = 0, andit is idempotent if elements of the form a1a2 span A. The following observationindicates the importance of these classes of algebras.

Let A be a ∗-algebra, and let AAA be the natural bimodule induced by left andright multiplications, equipped with the canonical inner products A〈a, b〉 =ab∗ and〈a, b〉A = a∗b.

Lemma 5.6. The bimodule AAA is a ∗- or strong equivalence bimodule if and onlyif A is nondegenerate and idempotent.


The proof is simple: idempotency is equivalent to the canonical inner productsbeing full and the actions by multiplication being strongly nondegenerate; non-degeneracy is equivalent to the inner products being nondegenerate. The innerproducts are completely positive by Example 3.9.

We therefore restrict ourselves to the class of nondegenerate and idempotent∗-algebras (which contains, in particular, all unital ∗-algebras). Within this class,∗-Morita equivalence is transitive [Ara 1999], hence it is an equivalence relation.We will show that the same holds for strong Morita equivalence.

The next result follows from arguments analogous to those in [Bursztyn andWaldmann 2001b, Lem. 3.1].

Lemma 5.7. Let A, B be nondegenerate and idempotent ∗-algebras, and let BEA

be a bimodule with inner products 〈 · , · 〉EA and B〈 · , · 〉E satisfying all the proper-

ties of Definition 5.1, except for nondegeneracy. Then their degeneracy spacescoincide, and the quotient bimodule E

/E⊥, with the induced inner products, is a

∗-equivalence bimodule. If 〈 · , · 〉EA and B〈 · , · 〉E are completely positive, then the

quotient bimodule is a strong equivalence bimodule.

As a result, within the class of nondegenerate and idempotent ∗-algebras, oneobtains a refinement of the internal tensor product ⊗ for equivalence bimodulestaking into account both inner products.

Lemma 5.8. Let A, B, C be nondegenerate and idempotent ∗-algebras and letBEA and CFB be ∗- (resp. strong) equivalence bimodules. Then the triple

(5–2) CFB ⊗B BEA :=(( CFB ⊗B BEA)

/( CFB ⊗B BEA)

⊥ , B〈 · , · 〉F⊗E, 〈 · , · 〉F⊗EA

)is a ∗- (resp. strong) equivalence bimodule.

Clearly ⊗ satisfies functoriality properties analogous to those of ⊗. CombiningLemmas 5.5, 5.6 and 5.8, we obtain:

Theorem 5.9. Strong Morita equivalence is an equivalence relation within theclass of nondegenerate and idempotent ∗-algebras over C.

5B. General properties. Let A and B be nondegenerate, idempotent ∗-algebras,and let 8 : A → B be a ∗-isomorphism. A simple check reveals that B, seen as an(A,B)-bimodule via

(5–3) a ·8 b · b1 =8(a)bb1

and equipped with the obvious inner products, is a strong equivalence bimodule.Hence ∗-isomorphism implies strong Morita equivalence.


On the other hand, [Ara 1999] shows that ∗-Morita equivalence (so also strongMorita equivalence) implies ∗-isomorphism of centers. As a result, for commuta-tive (nondegenerate and idempotent) ∗-algebras, strong and ∗-Morita equivalencescoincide with the notion of ∗-isomorphism.

Remark 5.10 (Finite-rank operators). Let (EA, 〈 · , · 〉EA) be an inner-product mod-

ule. The set of “finite-rank” operators on EA, denoted by F(EA), is the C-linearspan of operators θx,y ,

θx,y(z) := x · 〈y, z〉EA,

for x, y, z ∈ E. Note that θ∗x,y = θy,x and F(EA)⊆ B(EA) is an ideal.

Within the class of nondegenerate, idempotent ∗-algebras, an alternative de-scription of ∗-Morita equivalence is given as follows [Ara 1999]: if EA is a fullinner-product module so that EA ·A = EA, then F(E)EA is a ∗-equivalence bimodule,with F(EA)-valued inner product

(5–4) (x, y) 7→ θx,y .

On the other hand, if BEA is a ∗-equivalence bimodule, then the B-action on BEA

provides a natural ∗-isomorphism

(5–5) B ∼= F(EA).

Under this identification, the ∗-equivalence bimodule BEA becomes F(E)EA. Asa consequence, if EA is a pre-Hilbert module with EA · A = EA and (5–4) iscompletely positive, then F(E)EA is a strong equivalence bimodule.

The following is a standard example in Morita theory; see also [Bursztyn andWaldmann 2001a, Sect. 6].

Example 5.11 (Matrix algebras). Let A be a nondegenerate and idempotent ∗-algebra over C. We claim that A and Mn(A) are ∗- and strongly Morita equivalent.

First note that Cn is a strong (Mn(C),C)-equivalence bimodule. In fact, sinceF(Cn) = Mn(C) and Cn

· C = Cn , by Remark 5.10 it only remains to check that(5–4) is completely positive. But if x, y ∈ Cn , then we can write

θx,y = θx,e1θ∗

y,e1,

where e1 = (1, 0, . . . , 0)∈ Cn . So this inner product is of the form (3–8) (for m = 1and P1(x)= θx,e1), so it is completely positive.

By tensoring the equivalence bimodule AAA with Cn , it follows from Remark4.12 that the canonical inner products on An are completely positive. It then easilyfollows that An is a (Mn(A),A)-equivalence bimodule.

For unital ∗-algebras over C, it follows from the definitions that strong Moritaequivalence implies ∗-Morita equivalence, which in turn implies ring-theoretic


Morita equivalence. In particular, (∗- or strong) equivalence bimodules are finitelygenerated and projective with respect to both actions. Using the nondegeneracy ofinner products, their compatibility and fullness, one can verify this property directlyby checking that any ∗-equivalence bimodule admits a finite hermitian dual bases.As a consequence:

Corollary 5.12. If A, B and C are unital ∗-algebras and CFB and BEA are (∗-or strong) equivalence bimodules, then the inner product (4–4) on CFB ⊗B BEA isnondegenerate.

It follows that the quotient in (5–2) is irrelevant. This is always the case for (notnecessarily unital) C∗-algebras [Lance 1995, Prop. 4.5].

5C. Equivalence of categories of representations. It is shown in [Ara 1999] that∗-Morita equivalence implies equivalence of categories of (strongly nondegener-ate) representations on inner-product modules. We now recover this result andshow that an analogous statement holds for strong Morita equivalence, generalizing[Bursztyn and Waldmann 2001a, Thm. 5.10].

The next lemma follows from Lemmas 4.5 and 5.7.

Lemma 5.13. Let A, B, C and D be nondegenerate and idempotent ∗-algebras.Let CFB and BEA be ∗-equivalence bimodules, and let (H, π)∈ ∗-ModD(A). Thenthere are natural isomorphisms of inner-product bimodules:

( CFB ⊗B BEA) ⊗A AHD∼= CFB ⊗B ( BEA ⊗A AHD),(5–6)

AAA ⊗A AHD∼= AHD

∼= AHD ⊗D DDD.(5–7)

As a result, when CFB and BEA are strong equivalence bimodules, there is a natu-ral equivalence

(5–8) RF ◦ RE∼= RF⊗E.

Using the idempotency and nondegeneracy of A and B, one shows:

Lemma 5.14. Let BEA be a (∗- or strong) equivalence bimodule. If AEB is its con-jugate bimodule, then the following maps are (∗- or strong) equivalence bimoduleisomorphisms:

AEB ⊗B BEA → AAA, x ⊗B y 7→ 〈x, y〉EA,(5–9)

BEA ⊗A AEB → BBB, x ⊗A y 7→ B〈x, y〉E.(5–10)

Corollary 5.15. Let A, B and D be nondegenerate and idempotent ∗-algebras, andlet BEA be a strong equivalence bimodule. Then

(5–11) RE :∗-RepD(A)−→

∗-RepD(B)


is an equivalence of categories, with inverse given by RE.

Remark 5.16. Clearly, the functors SG satisfy a property analogous to (5–8); simi-larly to Corollary 5.15, an equivalence bimodule DGD′ establishes an equivalence ofcategories SG :

∗-RepD(A)→∗-RepD′(A). All these properties are direct analogs

of the previous constructions by replacing tensor products on the left by those onthe right.

Corollary 5.15 recovers the well-known theorem [Rieffel 1974b] on equivalenceof categories of nondegenerate ∗-representations of strongly Morita equivalent C∗-algebras on Hilbert spaces; see [Bursztyn and Waldmann 2001b; Ara 2001].

6. Picard groupoids

In this section, we introduce the Picard groupoid associated with strong Moritaequivalence, in analogy with the groupoid Pic [Benabou 1967] of invertible bimod-ules in ring-theoretic Morita theory [Morita 1958; Bass 1968]. (See [Landsman2001; Bursztyn and Weinstein 2004] for related constructions.)

6A. The strong Picard groupoid. Let ∗-Alg (resp. ∗-Alg+) be the category whoseobjects are nondegenerate and idempotent ∗-algebras over a fixed C, morphisms areisomorphism classes of inner-product (resp. pre-Hilbert) bimodules and composi-tion is internal tensor product (4–5). (The composition is associative by Lemma4.5.) We call an inner-product (resp. pre-Hilbert) bimodule BEA over A invertibleif its isomorphism class is invertible in ∗-Alg (resp. ∗-Alg+). Note that BEA isinvertible if and only if there exists an inner-product (resp. pre-Hilbert) bimoduleAE′

B over B together with isomorphisms

(6–1) AE′B ⊗B BEA

∼−→ AAA, BEA ⊗A AE′

B

∼−→ BBB.

Theorem 6.1. An inner-product (resp. pre-Hilbert) bimodule ( BEA, 〈 · , · 〉EA) is

invertible if and only if there exists a B-valued inner product B〈 · , · 〉E making( BEA, 〈 · , · 〉E

A, B〈 · , · 〉E) into a ∗-(resp. strong) equivalence bimodule. In partic-ular, ∗-(resp. strong) Morita equivalence coincides with the notion of isomorphismin ∗-Alg (resp. ∗-Alg+).

This is an algebraic version of a similar result in the framework of C∗-algebras[Landsman 2001; Schweizer 1999], which we will recover in Section 6B. We needthree main lemmas to prove the theorem.

Lemma 6.2. Let ( BEA, 〈 · , · 〉EA) be an invertible inner-product bimodule. Then

〈 · , · 〉EA is full and E·A = E. (By Remark 5.10, F(E)EA is a ∗-equivalence bimodule.)

Proof. Let AE′B be an inner-product bimodule such that (6–1) holds. The fullness

of 〈 · , · 〉EA is a simple consequence of the idempotency of A and the first isomor-

phism of (6–1).


For the second assertion, note that BEA · A ⊆ BEA is a (B,A) inner-productbimodule. Moreover, using the idempotency of A and the fact that A· AE′

B = AE′B,

it is simple to check that AE′B is still an inverse for BEA · A. By uniqueness of

inverses (up to isomorphism), we get BEA = BEA · A. �

Lemma 6.3. Let BEA be an invertible inner-product bimodule and let FB be aninner-product B-module. Then the natural map

SE : B(FB)−→ B(FB ⊗B BEA), T 7→ T ⊗B id,

is an isomorphism.

Proof. Let AE′B be as in (6–1). Then we have an induced map

SE′ : B(FB ⊗B BEA)−→ B((FB ⊗B BEA) ⊗A AE′B)∼= B(FB),

since (FB ⊗B BEA) ⊗A AE′B

∼= FB ⊗B ( BEA ⊗A AE′B)∼= FB ⊗B BBB

∼= FB. Onecan check that SE and SE′ are inverses of each other. �

The next result is an algebraic analog of [Lance 1995, Prop. 4.7].

Lemma 6.4. Let FB be an inner-product B-module so that FB · B = F. Let EA

be an inner-product A-module and π : B → B(EA) be a ∗-homomorphism so thatπ(B)⊆ F(EA). Then

SE(F(FB))⊆ F(FB ⊗B BEA).

Proof. Suppose y1, y2 ∈ FB and b ∈ B. Let y ⊗B x ∈ FB ⊗B BEA, and let θy1·b,y2 ∈

F(FB). Then

(6–2) SE(θy1·b,y2)(y ⊗B x)= θy1·b,y2 y ⊗B x = y1 · b〈y2, y〉FB ⊗B x

= y1 ⊗B π(b〈y2, y〉FB)x .

For each y ∈ FB, consider the map

ty : BEA −→ FB ⊗B BEA, ty(x)= y ⊗B x .

Then ty ∈ B( BEA, FB ⊗B BEA), with adjoint t∗y (y

′⊗B x ′)= π(〈y, y′

〉FB)x

′. We canrewrite (6–2) as

SE(θy1·b,y2)(y ⊗B x)= ty1π(b)t∗

y2(y ⊗B x).

Since π(b) ∈ F(EA), it follows that SE(θy1·b,y2) ∈ F(FB ⊗B BEA).For a general θy1,y2 , we use the condition that FB · B = FB to write y1 =∑kα=1 yα1 · bα and we repeat the argument above. �

We now prove Theorem 6.1 following [Landsman 2001; Schweizer 1999].


Proof. The fact that an equivalence bimodule BEA is invertible is a direct conse-quence of (5–9) and (5–10).

To prove the other direction, suppose that BEA is an invertible inner-productbimodule, with inverse AE′

B. By Lemma 6.3, we have two isomorphisms

(6–3) B(B)SE→ B(EA)

SE′

→ B(B),

whose composition is the identity. Recall that B = F(B)⊆ B(B). We claim that

SE′(F(EA))⊆ B.

Indeed, for x1, x2 ∈ BEA and b ∈B, we have SE′(θb·x1,x2)=bSE′(θx1,x2), which mustbe in B since B ⊂ B(B) is an ideal. For a general θx1,x2 , we use the conditionB ·E = E to write x1 =

∑kα=1 bα · xα1 and apply the same argument. By symmetry,

it then follows that

(6–4) SE(F(E′

B))⊆ A.

We now claim that

(6–5) SE(B)⊆ FA(E).

By Lemma 6.2, F(E′)EB is a ∗-equivalence bimodule. Consider its conjugate BE′F(E′).

Then

BBB∼= BE′

F(E′) ⊗F(E′) F(E′)EB,

and, as a consequence,

(6–6) BEA∼= BBB ⊗B BEA

∼= BE′F(E′) ⊗F(E′) ( F(E′)EB ⊗B BEA)∼= BE′

F(E′) ⊗F(E′) A,

where we regard A as a left F(E′)-module via (6–4).Since BE′

F(E′) is a ∗-equivalence bimodule, it follows that (see Remark 5.10)there is a natural identification

(6–7) B ∼= F(E′F(E′)).

Now consider the map

SA : B( BE′F(E′))−→ B( BE′

F(E′) ⊗F(E′) A).

By (6–4), F(E′B) acts on A via finite-rank operators; since BE′

F(E′) · F( F(E′)EB) =

BE′F(E′), we can apply Lemma 6.4 and use (6–6) and (6–7) to conclude that (6–5)

holds. We can restrict the isomorphisms in (6–3) to

BRE→ F(EA)

RE′

→ B,


which implies that RE′(F(EA))= B; since RE′ is injective (see Lemma 6.3), there isa natural ∗-isomorphism B ∼= F( BEA), so BEA is a ∗-equivalence bimodule, againby Remark 5.10.

If BEA and AE′B are pre-Hilbert bimodules, by uniqueness of inverses it follows

that

AEB∼= AE′

B

as pre-Hilbert bimodules, so the B-valued inner product on BEA must be com-pletely positive. So BEA is a strong equivalence bimodule. �

The invertible arrows in ∗-Alg (resp. ∗-Alg+) form a “large” groupoid Pic∗ (resp.Picstr), called the ∗-Picard groupoid (resp. strong Picard groupoid). By Theo-rem 6.1, orbits of Pic∗ (resp. Picstr) are ∗-Morita equivalence (resp. strong Moritaequivalence) classes and isotropy groups are isomorphism classes of self-∗-Moritaequivalences (resp. self-strong Morita equivalences), called ∗-Picard groups (resp.strong Picard groups).

If we restrict Pic, Pic∗ and Picstr to unital ∗-algebras over C, we obtain naturalgroupoid homomorphisms

(6–8) Picstr−→ Pic∗

−→ Pic,

covering the identity on the base. On morphisms, the first arrow “forgets” thecomplete positivity of inner products, while the second just picks the bimodulesand “forgets” all the extra structure.

In Section 7, we will discuss further conditions on unital ∗-algebras under whichthe canonical morphism

(6–9) Picstr−→ Pic

is injective and surjective.

Remark 6.5. The first arrow in (6–8) is generally not surjective since a bimodulemay have inner products with different signatures. For the same reason, the secondarrow is not injective in general.

6B. Strong Picard groupoids of C∗-algebras. Let C∗ be the category whose ob-jects are C∗-algebras and morphisms are isomorphism classes of Hilbert bimodules(see [Landsman 2001], for example); the composition is given by Rieffel’s internaltensor product in the C∗-algebraic sense. The groupoid of invertible morphisms inthis context will be denoted by Picstr

C∗ . The isotropy groups of PicstrC∗ are the Picard

groups of C∗-algebras as in [Brown et al. 1977].It is shown in [Schweizer 1999; Landsman 2001; Connes 1994] that Rieffel’s

notion of strong Morita equivalence of C∗-algebras coincides with the notion of


isomorphism in C∗. We will show how this result can be recovered from Theorem6.1.

For a C∗-algebra A, let P(A) be its minimal dense ideal, also referred to as itsPedersen ideal; see [Pedersen 1979]. Just as A itself, P(A) is nondegenerate andidempotent. If A and B are C∗-algebras, let BEA be a Hilbert bimodule (in theC∗-algebraic sense, see [Landsman 2001, Def. 3.2]) with inner product 〈 · , · 〉E

A,and consider the (P(B),P(A))-bimodule

P( BEA) := P(B) · BEA · P(A).

Lemma 6.6. The bimodule P( BEA), together with the restriction of 〈 · , · 〉EA, is a

pre-Hilbert (P(B),P(A))-bimodule (as in Definition 4.3).

Proof. It is clear that P(B) ·P( BEA)= P( BEA), and⟨P( BEA),P( BEA)

⟩EA

⊆ P(A),since 〈 · , · 〉E

A is A-linear and P(A)⊆ A is an ideal.For any n ∈ N, P(Mn(A)) = Mn(P(A)). So, by [Bursztyn and Waldmann

2001b, Lem. 3.2], A ∈ Mn(P(A))+ (in the algebraic sense of Section 2) if and

only if A ∈ Mn(A)+. Since 〈 · , · 〉E

A is completely positive, so is its restriction toP( BEA) taking values in P(A). �

The next example shows that P( BEA) ⊆ BEA · P(A) is essential to guaranteethat the restriction of the inner product takes values in P(A).

Example 6.7. Let X be a locally compact Hausdorff space, and consider A =

C∞(X), the algebra of continuous functions vanishing at ∞, and B = C. ThenE = C∞(X) is naturally a (B,A)-Hilbert bimodule. Since B is unital, B = P(B)

and P(B) · E = E. But P(A) = C0(X) is the algebra of compactly supportedfunctions. If X is not compact, then 〈E,E〉

EA * P(A).

Let A,B and C be C∗-algebras. For Hilbert bimodules CFB and BEA, we denotetheir Rieffel internal tensor product in the C∗-algebraic sense by CFB⊗B BEA; see[Rieffel 1974a; 1974b; Raeburn and Williams 1998]. A direct verification gives:

Lemma 6.8. There is a canonical isomorphism P( CFB⊗B BEA) ∼= P( CFB) ⊗B

P( BEA).

Write ∗-Alg(+) for either ∗-Alg or ∗-Alg+. With some abuse of notation, it followsfrom Lemmas 6.6 and 6.8 that we can define a functor

(6–10) P : C∗−→

∗-Alg(+)

as follows: on objects, A 7→ P(A); on morphisms, P([ BEA]) = [P( BEA)]. Here[ ] denotes the isomorphism class of a (pre-)Hilbert bimodule.

Any pre-Hilbert bimodule of the form E = P( BEA) must satisfy E · P(A)= E.So the maps on morphisms induced by (6–10), P : Mor(A,B)→ Mor(PA,PB),


are not surjective in general. However, as we will see, the situation changes if werestrict P to morphisms which are invertible.

Our main analytical tool is the next result; see [Ara 2001].

Proposition 6.9. Let A and B be C∗-algebras.

(i) If PBEPA

is a strong (or ∗-)equivalence bimodule (as in Definition 5.1), then itcan be completed to a C∗-algebraic strong equivalence bimodule BEA in sucha way that P( BEA)∼= PB

EPA.

(ii) If BFA and BEA are C∗-algebraic strong equivalence bimodules such thatP( BFA)∼= P( BEA), then BFA

∼= BEA.

Proof. The proof of (i) follows from the results in [Ara 2001]. Note that EP(A)

can be completed to a full Hilbert A-module EA (see [Lance 1995, p. 5]), so that

K(EA)EA is a strong equivalence bimodule. Here K(EA) denotes the “compact”

operators on EA. Note that E is naturally an A-module and sits in EA as a dense A-submodule. So P(B)= F(EA) is dense in K(EA), and B is naturally ∗-isomorphicto K(EA). So BEA is a C∗-algebraic strong equivalence bimodule. It follows from[Ara 2001, Thm. 2.4] that any (P(B),P(A))-∗-equivalence bimodule is already astrong equivalence bimodule, so the same results hold.

It follows from [Ara 2001] that

P(B) · BEA = BEA · P(A)= P(B) · BEA · P(A).

Since P(B) · PBEPA

· P(A) = PBEPA

, we have PBEPA

⊆ P(B) · BEA · P(A). Onthe other hand, since P(B) ⊂ B(EA) is an ideal, it follows that E ⊂ E is B(EA)-invariant. By [Ara 2001, Prop. 1.5], P(B) · BEA · P(A) ⊆ E. This implies thatP( BEA)= PB

EPA.

Part (ii) follows from the fact that BEA is a completion of P( BEA) and any twocompletions must be isomorphic. �

Corollary 6.10. A Hilbert bimodule BEA is invertible in C∗ if and only if thereexists a B-valued inner product B〈 · , · 〉E so that ( BEA, B〈 · , · 〉E, 〈 · , · 〉E

A) is a (C∗-algebraic) strong equivalence bimodule. Thus, in particular, two C∗-algebras arestrongly Morita equivalent if and only if they are isomorphic in C∗.

Proof. If BEA is invertible in C∗, then P( BEA) is invertible in ∗-Alg+. By Theorem6.1, there exists a B-valued inner product making P( BEA) into an equivalencebimodule. By Proposition 6.9(i), we can complete it to a C∗-algebraic strongequivalence bimodule, isomorphic to BEA as a Hilbert bimodule. �

Corollary 6.11. For C∗-algebras A and B,

(6–11) P : PicstrC∗(B,A)−→ Picstr(P(B),P(A))


is a bijection. As a result, PicstrC∗ is equivalent to Picstr (or Pic∗) restricted to Peder-

sen ideals.

It follows that the entire strong Morita theory of C∗-algebras is encoded in thealgebraic Picstr. Note that, for unital C∗-algebras, P is just the identity on objects.

7. Strong versus ring-theoretic Picard groupoids

It is shown in [Beer 1982] that unital C∗-algebras are strongly Morita equivalentif and only if they are Morita equivalent as rings. In [Bursztyn and Waldmann2002], we have shown that the same is true for hermitian star products. In termsof Picard groupoids, these results mean that Picstr and Pic, restricted to unital C∗-algebras or to hermitian star products, have the same orbits. In this section, westudy the morphism Picstr

→Pic restricted to unital ∗-algebras satisfying additionalproperties, recovering and refining these results in a unified way.

7A. A restricted class of unital ∗-algebras. We consider algebraic conditions thatcapture some important features of the functional calculus of C∗-algebras. Let A

be a unital ∗-algebra over C. The first property is

(I) For all n ∈ N and A ∈ Mn(A), 1+ A∗ A is invertible.

As a first remark we see that (I) also implies that elements of the form

1+∑k

r=1 A∗r Ar

are invertible in Mn(A), simply by applying (I) to Mnk(A). The relevance of thisproperty is illustrated by the following result [Kaplansky 1968, Thm. 26]:

Lemma 7.1. Suppose A satisfies (I). Then any idempotent e = e2∈ Mn(A) is

equivalent to a projection P = P2= P∗

∈ Mn(A).

We also need the following property.

(II) For all n ∈ N, let Pα ∈ Mn(A) be pairwise orthogonal projections, i.e. PαPβ =

δαβ Pα, with 1 =∑

α Pα and let H ∈ Mn(A)+ be invertible. If [H, Pα] = 0,

then there exists an invertible U ∈ Mn(A) with H = U∗U and [Pα,U ] = 0.

Most of our results will follow from a condition slightly weaker than (II):

(II−) For all n ∈ N, invertible H ∈ Mn(A)+, and projection P with [P, H ] = 0,

there exists a U ∈ Mn(A) with H = U∗U and [P,U ] = 0.

On the other hand, our main examples satisfy a stronger version of (II):

(II+) For all n ∈N and H ∈ Mn(A)+ invertible there exists an invertible U ∈ Mn(A)

such that H = U∗U , and if [H, P] = 0 for a projection P then [U, P] = 0.


Any unital C∗-algebra fulfills (I) and (II+) by their functional calculus. In Sec-tion 8C we show that the same holds for hermitian star products. The importanceof condition (II) and its variants lies in the next result.

Lemma 7.2. Let A satisfy (II−) and let P = P2= P∗

∈ Mn(A). Then any com-pletely positive and strongly nondegenerate A-valued inner product on PAn isisometric to the canonical one.

Proof. Given such inner product 〈 · , · 〉′ on PAn , we extend it to the free module An

by taking (1− P)An as orthogonal complement with the canonical inner productof An restricted to it. Then the result follows from Lemma 3.12 and the fact thatthe isometry on An commutes with P . �

Let R be an arbitrary unital ring. An idempotent e = (ei j ) ∈ Mn(R) is calledfull if the ideal in R generated by ei j coincides with R. One of the main resultsof Morita theory for rings [Bass 1968] is that two unital rings R and S are Moritaequivalent if and only if S ∼=eMn(R)e for some full idempotent e. The next theoremis an analogous result for strong Morita equivalence.

For a projection P ∈ Mn(A), we consider PAn equipped with its canonicalcompletely positive inner product. Note that P is full if and only if this innerproduct is full in the sense of (5–1).

Theorem 7.3. Let A, B be unital ∗-algebras and let ( BEA, B〈 · , · 〉E, 〈 · , · 〉EA) be a

∗-equivalence bimodule such that 〈 · , · 〉EA is completely positive. If A satisfies (I)

and (II−) then:

(1) There exists a full projection P = P2= P∗

∈ Mn(A) such that EA is isomet-rically isomorphic to PAn as a right A-module.

(2) B is ∗-isomorphic to P Mn(A)P via the left action on EA and the B-valuedinner product is, under this isomorphism, given by the canonical P Mn(A)P-valued inner product on PAn .

(3) B〈 · , · 〉E is completely positive and hence BEA is a strong equivalence bimod-ule.

Conversely, if P is a full projection, then P Mn(A)P is strongly Morita equivalentto A via PAn .

Proof. We know that EA is finitely generated and projective. By (I) we can find aprojection P with EA

∼= PAn and by (II−) we can choose the isomorphism to beisometric to the canonical inner product, according to Lemma 7.2, proving the firststatement.

Since B〈 · , · 〉E is full, the left action map is an injective ∗-homomorphism of B

into B(EA). By compatibility, B〈 · , · 〉E has to be the canonical one and again by


fullness we see that B is ∗-isomorphic to B(EA)∼= P Mn(A)P , proving the secondstatement.

Since 〈 · , · 〉EA is full we find Pxr , Pyr ∈ PAn with 1A =

∑kr=1〈Pxr , Pyr 〉. Since

1A = 1∗

A we have∑r 〈Pxr + Pyr , Pxr + Pyr 〉 = 1A + 1A +

∑r 〈Pxr , Pxr 〉 +

∑r 〈Pyr , Pyr 〉.

By (I) and (II−) we find an invertible U ∈ A such that for Pzr = P(xr + yr )U−1∈

PAn we have 1A =∑

r 〈Pzr , Pzr 〉. By compatibility, we get

(7–1) B〈Px, Py〉E=∑

r B〈Px, Pzr 〉E

B〈Pzr , Py〉E;

the complete positivity of B〈 · , · 〉E now follows from Remark 3.10. This also showsthe last statement. �

We use Theorem 7.3 to show that condition (I) and (II) are natural from a Morita-theoretic point of view.

Proposition 7.4. Conditions (I) and (II+) (resp. (II)), together, are strongly Moritainvariant.

Proof. Assume that A satisfies (I) and (II−) and B is strongly Morita equivalent toA. Then B ∼= P Mn(A)P for some full projection P . If B ∈ B, then 1B + B∗B,viewed as element in P Mn(A)P , can be extended “block-diagonally” to an elementof the form

(7–2) 1Mn(A) + A∗ A

by addition of 1Mn(A) − P . By (I), (7–2) has an inverse in Mn(A). By (II−), theinverse is again block-diagonal and hence gives an inverse of 1B + B∗B. Passingfrom Mn(A) to Mnm(A), one obtains the invertibility of 1Mm(B) + B∗B for B ∈

Mm(B). Hence B satisfies (I).Assume that A satisfies (II+), and let H ∈ B+ be invertible. Then

H + (1Mn(A) − P) ∈ Mn(A)+

is still positive and invertible. So there is an invertible V ∈ Mn(A) with H +

(1Mn(A)− P)= V ∗V , commuting with P since H + (1Mn(A)− P) commutes withP . Thus U = PV P satisfies U∗U = H . Moreover, if Q ∈ B is a projection with[H, Q] = 0, then P Q = Q = Q P and hence Q commutes with H + (1Mn(A)− P).Thus V commutes with Q, and hence U commutes with Q as well. For Mm(B),the reasoning is analogous. So B satisfies (II+).

An analogous but simpler argument shows the same result for (II). �


7B. From Picstr to Pic. Consider the groupoid morphism

(7–3) Picstr−→ Pic

from the strong Picard groupoid to the (ring-theoretic) Picard groupoid. The nextresult follows from Theorem 7.3.

Theorem 7.5. Within the class of unital ∗-algebras satisfying (I) and (II−), thegroupoid morphism (7–3) is injective.

For surjectivity, first note that if we define the Hermitian K0-group of a ∗-algebraA as the Grothendieck group K H

0 (A) of the semi-group of isomorphism classes offinitely generated projective pre-Hilbert modules over A equipped with stronglynondegenerate inner products, then Lemmas 7.1 and 7.2 imply that if A satisfies(I) and (II−), then the natural group homomorphism K H

0 (A)→ K0(A) (forgettinginner products) is an isomorphism. For Picard groupoids, however, we will seethat (7–3) is not generally surjective, even if (I) and (II) hold. In order to discussthis surjectivity problem, we consider pairs of ∗-algebras satisfying the followingrigidity property:

(III) Let A and B be unital ∗-algebras, let P ∈ Mn(A) be a projection, and con-sider the ∗-algebra P Mn(A)P . If B and P Mn(A)P are isomorphic as unitalalgebras, then they are ∗-isomorphic.

The following are the motivating examples.

Examples 7.6. (1) For unital C∗-algebras, condition (III) is always satisfied: If A

is a C∗-algebra, then so is P Mn(A)P , and (III) follows from the fact that twoC∗-algebras which are isomorphic as algebras must be ∗-isomorphic [Sakai1971, Thm. 4.1.20].

(2) Another class of unital ∗-algebras satisfying (III) is that of hermitian starproducts on a Poisson manifold M ; see Section 8. In this case, condition (III)follows from the more general fact that two equivalent star products which arecompatible with involutions of the form f 7→ f + o(λ) must be ∗-equivalent;see [Bursztyn and Waldmann 2002, Lem. 5].

For unital algebras A and B, let us consider the action of the automorphismgroup Aut(B) on the set of morphisms Pic(B,A) by

(7–4) (8, [E]) 7→ [8E];

here E is a (B,A)-equivalence bimodule (in the ring-theoretic sense), 8∈ Aut(B)and 8E coincides with E as a C-module, but its (B,A)-bimodule structure is givenby

b ·8 x · a :=8(b) · x · a;

see for example [Bass 1968; Bursztyn and Waldmann 2004a].


Proposition 7.7. If A and B are unital ∗-algebras satisfying (III), and if A satisfies(I) and (II−), then the composed map

(7–5) Picstr(B,A)−→ Pic(B,A)−→ Pic(B,A)/

Aut(B)

is onto.

Proof. Let BEA be an equivalence bimodule for ordinary Morita equivalence. Weknow that EA

∼=eAn , as right A-modules, for some full idempotent e=e2∈ Mn(A),

and B ∼= eMn(A)e as associative algebras via the left action.By properties (I) and (II−), we can replace e by a projection P = P2

= P∗

and consider the canonical A-valued inner product on PAn . Then P Mn(A)P andA are strongly Morita equivalent via PAn; see Theorem 7.3. The identificationB ∼= P Mn(A)P induces a ∗-involution † on B, possibly different from the originalone. By assumption, there exists a ∗-isomorphism

8 : (B, ∗)−→ (B, †)

in such a way that 8E becomes a strong equivalence bimodule. �

Corollary 7.8. Within a class of unital ∗-algebras satisfying (I), (II) and (III), ring-theoretic Morita equivalence implies strong Morita equivalence (so the two notionscoincide).

Proposition 7.7 is an algebraic refinement of Beer’s result for unital C∗-algebras[Beer 1982], which is recovered by Corollary 7.8.

The question to be addressed is when (7–3) is surjective, and not only surjectivemodulo automorphisms. The obstruction for surjectivity is expressed in the nextcondition.

(IV) For any8∈Aut(A) there is an invertible U ∈A such that8∗8−1=Ad(U∗U ),

where 8∗(a)=8(a∗)∗.

Lemma 7.9. Assume that a unital ∗-algebra A satisfies (I) and (II−), and let BEA

be a ring-theoretic equivalence bimodule whose class [ BEA] ∈ Pic(B,A) is in theimage of (7–3). Then its entire Aut(B)-orbit is in the image of (7–3) if and only ifB satisfies (IV).

Proof. If the isomorphism class of BEA is in the image of (7–3), then there is a fullcompletely positive A-valued inner product 〈 · , · 〉E

A which is uniquely determinedup to isometry by the right A-module structure.

If 8 ∈ Aut(B), then [8E] is in the image of (7–3) if and only if there is an A-valued inner product 〈 · , · 〉′, necessarily isometric to 〈 · , · 〉E

A by Lemma 7.2, whichis compatible with the B-action modified by 8. In this case, the B-valued inner


product is determined by compatibility, and its complete positivity follows fromTheorem 7.3. Since there exists an invertible U ∈ BA(E)= B such that

〈x, y〉′= 〈U · x,U · y〉

EA,

condition (IV) easily follows from the nondegeneracy of 〈 · , · 〉EA. �

Corollary 7.10. Let A and B be unital ∗-algebras satisfying (III), and supposethat A satisfies properties (I) and (II−). Then the first map in (7–5) is surjective ifand only if B satisfies (IV).

Corollary 7.11. Within a class of unital ∗-algebras satisfying (I), (II−) and (III),property (IV) is strongly Morita invariant.

Example 7.12 (The case of C∗-algebras). For a unital C∗-algebra A, any automor-phism 8 ∈ Aut(A) can be uniquely decomposed as

(7–6) 8= eiD◦9,

where 9 is a ∗-automorphism and D is a ∗-derivation, i.e., a derivation withD(a∗) = D(a)∗; see [Okayasu 1974, Thm. 7.1] and [Sakai 1971, Cor. 4.1.21]. Inthis case, (IV) is satisfied if and only if, for any ∗-derivation D, the automorphismeiD is inner.

We discuss some concrete examples. If A is a simple C∗-algebra or a W ∗-algebra, then any automorphism is inner; see [Sakai 1971, Thm. 4.1.19]. So (IV)is automatically satisfied, and (7–3) is surjective.

In general, however, there may be automorphisms 8 with 8∗= 8−1 such that

82 is not inner, in which case (7–3) is not surjective. For example, consider thecompact operators K(H) on a Hilbert space H with countable Hilbert basis en .Define A = A∗

∈ B(H) by

Ae2n = 2e2n and Ae2n+1 = e2n+1.

Then Ad(A) induces an automorphism 8 of K(H)⊕C1 which satisfies 8∗=8−1

but whose square is not inner: clearly Ad(A)2 = Ad(A2) and there is no B ∈

K(H)⊕ C1 with Ad(A2)= Ad(B∗B).

8. Hermitian deformation quantization

We now show that, just as C∗-algebras, hermitian star products can be treated inthe framework of Section 7. The key observation is that the properties consideredin Section 7 are rigid under deformation quantization.


8A. Hermitian and positive deformations of ∗-algebras. Let A be a ∗-algebraover C. Let A = (A[[λ]], ?) be an associative deformation of A, in the sense of[Gerstenhaber 1964]. We call this deformation hermitian if

(a1 ? a2)∗= a∗

2 ? a∗

1 ,

for all a1, a2 ∈ A. In this case, ∗ can be extended to a ∗-involution making A into a∗-algebra over C[[λ]]. Note that C[[λ]] = R[[λ]](i), and R[[λ]] has a natural orderinginduced from R, see Section 1, so all the notions of positivity of Section 2 makesense for A. We assume λ to be real, so λ= λ > 0.

If ω =∑

∞

r=0 λrωr : A[[λ]] → C[[λ]] is a positive C[[λ]]-linear functional with

respect to ?, then its classical limit ω0 : A → C is a positive C-linear functionalon A. We say that a hermitian deformation A = (A[[λ]], ?) is positive [Bursztynand Waldmann 2000b, Def. 4.1] if every positive linear functional on A can bedeformed into a positive linear functional of A. In the spirit of complete positivity,we call a deformation A completely positive if, for all n ∈ N, the ∗-algebras Mn(A)

are positive deformations of Mn(A). We remark that not all hermitian deformationsare positive [Bursztyn and Waldmann 2004b].

In the following, we shall consider unital ∗-algebras and assume that hermitiandeformations preserve the units.

8B. Rigidity of properties (I) and (II). The next observation is a direct conse-quence of the definitions.

Lemma 8.1. Let A be a positive deformation of A. If a = a +o(λ) ∈ A is positive,then its classical limit a ∈ A is also positive.

A property of a ∗-algebra A is said to be rigid under a certain type of deformationif any such deformation satisfies the same property. Clearly, property (I) is rigidunder hermitian deformations. More interestingly:

Proposition 8.2. Property (II) is rigid under completely positive deformations.

Proof. Let H = H + o(λ) ∈ Mn(A)+ be positive and invertible, and let Pα =

Pα + o(λ) ∈ Mn(A) be pairwise orthogonal projections satisfying∑

α Pα = 1

and [H, Pα]? = 0. By Lemma 8.1, H ∈ Mn(A) is positive and invertible. Since[Pα, H ] = 0, by (II) there exists an invertible U ∈ Mn(A) with H = U∗U and[Pα,U ] = 0. In particular, PαU Pα ∈ PαMn(A)Pα is invertible, with inversePαU−1 Pα; here we consider PαMn(A)Pα as a unital ∗-algebra with unit Pα asbefore. Hence

(8–1) PαH Pα = PαU∗ PαPαU Pα.

But Pα ? Mn(A) ? Pα induces a hermitian deformation ?α of PαMn(A)Pα, so wecan apply [Bursztyn and Waldmann 2000a, Lem. 2.1] to deform (8–1), i.e., there


exists an invertible Uα ∈ Pα ?Mn(A) ? Pα such that

(8–2) Pα ? H ? Pα = Pα ?U∗

α ? Pα ? Pα ?Uα ? Pα.

If we set U =∑

α Uα, then it is easy to check that U commutes with the Pα, isinvertible and H = U∗ ?U . �

By only considering the projections P and (1− P), one can show that property(II−) is rigid under completely positive deformations as well.

As a consequence, any completely positive deformation of a ∗-algebra A satis-fying (I) and (II) (or (II−)) also satisfies these properties and those resulting fromthem, as discussed in Section 7.

8C. Hermitian star products. A star product [Bayen et al. 1978a; 1978b] on aPoisson manifold (M, {·, ·}) is a formal deformation ? of C∞(M),

f ? g = f g +

∞∑r=1

λr Cr ( f, g),

for which each Cr is a bidifferential operator and

C1( f, g)− C1(g, f )= i{ f, g}.

Following Section 8A, a star product is hermitian if ( f ? g)= g ? f .In [Bursztyn and Waldmann 2000b, Prop. 5.1], we proved that any hermitian

star product on a symplectic manifold is a positive deformation. This turns out tohold much more generally.

Theorem 8.3. Any hermitian star product on a Poisson manifold is a completelypositive deformation.

The proof consists of showing that any hermitian star product can be realized asa subalgebra of a formal Weyl algebra, and then use the results in the symplecticcase [Bursztyn and Waldmann 2000b], see [Bursztyn and Waldmann 2004b].

Since C∞(M) satisfies (I) and (II), we have

Corollary 8.4. Hermitian star products on Poisson manifolds satisfy properties (I)and (II).

Corollary 8.5. Let E → M be a hermitian vector bundle. Then any hermitiandeformation of 0∞(End(E)) satisfies (I) and (II).

Proof. Any such deformation is strongly Morita equivalent to some hermitian starproduct on M ; see [Bursztyn and Waldmann 2002; 2000a]. Thus the result followsfrom Proposition 7.4. �


Knowing that hermitian star products are completely positive deformations, wecan use the star exponential to show that they satisfy a property which is muchstronger than (II).

Proposition 8.6. Let ? be a hermitian star product on M . Then any positiveinvertible H ∈ Mn(C∞(M)[[λ]])+ has a unique positive invertible ?-square root?√

H such that [?√

H , A]? = 0 if and only if [H, A]? = 0, A ∈ Mn(C∞(M)[[λ]]). Inparticular, ? satisfies (II+).

Proof. If H = H0 + o(λ) then, by Lemma 8.1, H0 is positive in Mn(C∞(M)) andinvertible. This implies that H0 has a unique real logarithm ln(H0)∈ Mn(C∞(M)).Using the star exponential as in [Bursztyn and Waldmann 2002; 2004a], extendedto matrix-valued functions, we conclude that there exists a unique real star loga-rithm Ln H = ln H0 + o(λ) of H , whence Exp Ln H = H . It follows that ?

√H =

Exp( 1

2 Ln H)

has the desired property. �

This shows that many important features of the functional calculus of C∗-algebrasare present in formal deformation quantization.

8D. The strong Picard groupoid for star products. Since hermitian star productssatisfy (I), (II+) and (III), it follows that Theorem 7.5 and Proposition 7.7 hold.

Corollary 8.7. For hermitian star products, Picstr and Pic have the same orbits andthe canonical morphism Picstr

→ Pic is injective.

Corollary 8.7 recovers [Bursztyn and Waldmann 2002, Thm. 2]. The orbits andisotropy groups of the Picard groupoid in deformation quantization were studiedin [Bursztyn and Waldmann 2002; 2004a; Jurco et al. 2002].

The next result reveals an interesting similarity between the structure of the au-tomorphism group of C∗-algebras and hermitian star products; see Example 7.12.

Proposition 8.8. Let ? be a hermitian star product on a Poisson manifold M , andlet 8 ∈ Aut(?) be an automorphism of ?. Then there exists a unique ∗-derivationD and a unique ∗-automorphism 9 such that

(8–3) 8= eiλD◦9.

Proof. Writing 8 =∑

∞

r=0 λr8r , we know that 80 = ϕ∗ is the pull-back by some

Poisson diffeomorphism ϕ : M → M . In particular, 80( f )=80( f ).Define a new star product ?′ by f ?′ g = ϕ∗(ϕ∗ f ? ϕ∗g). Then ?′ is hermitian

and ∗-isomorphic to ? via ϕ∗. If we write 8 = T ◦ ϕ∗, then T = id +o(λ). Hence? and ?′ are equivalent via T .

By [Bursztyn and Waldmann 2002, Cor. 4], there exists a ∗-equivalence T be-tween ? and ?′, so 9(1)

= ϕ∗◦ T is a ∗-automorphism of ? deforming ϕ∗. By

[Bursztyn and Waldmann 2002, Lem. 5], there is a unique derivation D(1) so that


8 = eiλD(1)◦9(1), and we can write D(1)

= D(1)1 + iD(1)

2 , where each D(1)i is a ∗-

derivation. Now the Baker–Campbell–Hausdorff formula defines a derivation D(2)

by eiλD◦ eλD(1)

2 = eiλD(2), in such a way that the imaginary part of D(2) is of order

λ. By induction, we can split off the ∗-automorphisms eλD(k)2 to obtain (8–3). A

simple computation shows the uniqueness of this decomposition. �

Using this result, we proceed in total analogy with the case of C∗-algebras. Aderivation of a star product ? is quasi-inner if it is of the form D = (i/λ) ad H forsome H ∈ C∞(M)[[λ]].

Theorem 8.9. Let ?, ?′ be Morita equivalent hermitian star products. Then

(8–4) Picstr(?, ?′)−→ Pic(?, ?′)

is a bijection if and only if all derivations of ? are quasi-inner.

Proof. We know that (8–4) is injective, and it is surjective if and only if anyautomorphism 8 of ? satisfies

8 ◦8−1= Ad(U ?U )

for some invertible function U ; see Corollary 7.10. Using (8–3), this is equivalentto the condition that, for any ∗-derivation D, e−2iλD

= Ad(U ?U ).Since U ?U = U 0U0 + o(λ) for some invertible U0, we can use the unique real

star logarithm Ln H of H = U ?U to write

e−2iλD= Ad(Exp Ln H)= ead Ln H .

Hence we have the equivalent condition D = (i/λ) ad(1

2 Ln H)

for any ∗-derivation.Since any derivation can be decomposed into real and imaginary parts, each beinga ∗-derivation, the statement follows. �

If ? is a hermitian star product for which Poisson derivations can be deformedinto ?-derivations in such a way that hamiltonian vector fields correspond to quasi-inner derivations, then ?-derivations modulo quasi-inner derivations are in bijectionwith formal power series with coefficients in the first Poisson cohomology; see, forexample, [Gutt and Rawnsley 1999; Bursztyn and Waldmann 2004a]. In this case,(8–4) is an isomorphism if and only if the first Poisson cohomology group van-ishes. We recall that any Poisson manifold admits star products with this property[Cattaneo et al. 2002], and any symplectic star product is of this type.

Corollary 8.10. If ? is a hermitian star product on a symplectic manifold M , then(8–4) is an isomorphism if and only if H1

dR(M,C)= {0}.


Acknowledgments

We thank Martin Bordemann for many valuable discussions. We also thank S.Jansen, N. Landsman, I. Moerdijk, R. Nest, and A. Weinstein for useful commentsand remarks. Bursztyn thanks Freiburg University and IPAM–UCLA for their hos-pitality while part of this work was being done.

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Received April 6, 2004.

HENRIQUE BURSZTYN

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF TORONTO

100 ST. GEORGE STREET

TORONTO, ONTARIO, M5S 3G3CANADA

[email protected]

STEFAN WALDMANN

FAKULTAT FUR MATHEMATIK UND PHYSIK

ALBERT-LUDWIGS-UNIVERSITAT FREIBURG

PHYSIKALISCHES INSTITUT

HERMANN HERDER STRASSE 3D 79104 FREIBURG

GERMANY

[email protected]


INTEGER POINTS ON ELLIPTIC CURVES

WEN-CHEN CHI, KING FAI LAI AND KI-SENG TAN

We study Lang’s conjecture on the number of S-integer points on an ellip-tic curve over a number field. We improve the exponent of the bound ofGross and Silverman from quadratic to linear by using the S-unit equationmethod of Evertse and a formula on 2-division points.

1. Introduction

Let E be an elliptic curve defined over an algebraic number field k of degree d . Fora finite set S of places of k containing all the archimedean ones, we denote the ringof S-integers of k by OS . Serge Lang conjectured that if the Weierstrass equationof E is quasiminimal, then the cardinality of the set E(OS) of OS-integer points ofE should be bounded in terms of the field k, the cardinality of S and the rank ofthe group E(k) of k-rational points of E [Lang 1978, p. 140]. Silverman [1987]proved Lang’s conjecture when E has integral j-invariant. In general, if j (E) isnonintegral for at most δ places of k, then a bound was also given with δ involved.However he did not compute the constants involved. Gross and Silverman [1995]used Roth’s theorem to obtain an explicit bound. To state their theorem, let uswrite the Weierstrass equation of the elliptic curve E as

(1–1) Y 2= X3

+ AX + B,

where A,B ∈ OS . Put 1 = 4A3+ 27B2. Write j (E) for the j-invariant of E .

Let Dk and Rk be the discriminant and the regulator of k. Let Mk be the set of allplaces of k. For a place v ∈ Mk , let kv be the completion of k at v and let | |v besuch that, for z ∈ Q,

|z|v = |z|[kv :Qp]/[k:Q]

p ,

MSC2000: primary 11D45; secondary 11G05, 14K12.Keywords: elliptic curves, S-integers, integer points, S-unit equations, 2-division points, Lang’s

conjecture.Chi and Tan were supported in part by the National Science Council of Taiwan, grants NSC91-2115-M-003-006 and NSC89-2115-M-002-003 respectively.

237

238 WEN-CHEN CHI, KING FAI LAI AND KI-SENG TAN

where p is the place of Q lying under v and | |p is the usual absolute value. Weuse hk to denote the multiplicative height. Namely, for x ∈ k

hk(x)=

∏v∈Mk

max(|x |v, 1).

We shall write s for the cardinality of the set S.

Theorem 1.1 [1995]. Suppose that (1–1) is quasiminimal and that

6d(60d2 log 6d)d( 2√

3

)d(d−1)/2· max(Rk, log |Dk |, 1).

is at most

max{log hk( j (E)), log |Normk/Q(1)|

}.

Then

#E(OS)≤ 2 · 1011· d · δ3d

· (32 · 109)rδ+s .

In this paper, we take a completely different approach. By using a formula on2-division points from [2002], we associate to an S-integer point an unit equationover an extension of k. Then we use the machinery developed by J.-H. Evertse[1984] to obtain a quantitative bound for the number of S-integer points. Let DE/k

be the ideal of the minimal discriminant of E/k. Then we have

(1–2) (1)= DE/k ·

∏v

P12χvv ,

where Pv is the prime ideal corresponding to the place v and χv ∈ Z. For v ∈ S,χv ≥ 0. We factor the cubic over the algebraic closure k of k as

X3+ AX + B = (X −α)(X −β)(X − γ ).

Let k1 = k(α, β, γ ) and m = [k1 : k]. Further, let Mk,0 be the set of all nonar-chimedean places in k.

Definition 1.2. Let w be a nonarchimedean place over a field extension K/k1. Ifthe valuations w(α − β), w(β − γ ), w(γ − α) are all equal, we say that E hasG-type reduction at w; otherwise, we say that E has M-type reduction at w.

In fact, ifw′ is another place of K such that bothw andw′ are sitting over a placev ∈ Mk,0, then the reductions of E at w and w′ are of the same type. Therefore,we will say that at v, the reduction of E is also of that type. Furthermore, in thecase where v(2)= 0, E has G-type reduction if and only if it has good or potentialgood reduction (see Lemma 3.1).

INTEGER POINTS ON ELLIPTIC CURVES 239

Define

S0 = {v ∈ Mk,0 \ S | v(2)= 0, χv = 0, v(1) > 0, v( j (E))≥ 0},

S1 = {v ∈ Mk,0 | χv > 0, v( j (E))≥ 0},

Sm = {v ∈ Mk,0 | E has M-type reduction at v},

S′= S \ (S0 ∪ S1 ∪ Sm).

Let s1, sm , s ′ be the cardinality of S1, Sm , S′. Then sm is at most δ+ d .With the notations above, we can now state our main result.

Theorem 1.3.

#E(OS)≤ 11 × 71.64r+2.27(s′+s1)+3.7sm+10.3md .

Note that we do not require the equation (1–1) to be quasiminimal. If we didso, then, by [Silverman 1984, p. 238], we would have∣∣∣∣Normk/Q

∏v∈S1

Pχv∣∣∣∣≤ |Dk |

6,

and hences1 ≤ 6 log |Dk |.

The exponent in the Gross–Silverman bound is quadratic in δ and r , while oursis linear, and our constants are smaller. Also, if the ABC Conjecture holds, ourmethod can be applied to get a bound only in terms of r and k, in which theexponent is linear in s and r and differs from that obtained in [Hindry and Silverman1988]. In fact, this has been achieved in [Chi et al. 2004] for the case where k isa function field of characteristic zero. Also, the method can be modified to boundthe number of integer solutions to Y n

= F(X); see [Chi et al. ≥ 2006].

2. A formula for 2-division points

The following result can be proved by straightforward calculations. For details,see [Tan 2002] or [Chi et al. 2004, Section 2.2].

Lemma 2.1. In the notations preceding Theorem 1.3 a point P = (a, b) ∈ E(k)determines an extension

K = k1(√

a −α,√

a −β,√

a − γ )

depending only on the class [P] ∈ E(k)/2E(k). Given a choice of signs for√

a−α,√

a−β, and√

a−γ such that

b =√

a−α√

a−β√

a−γ ,


the point Q := ( f, g) ∈ E(K ) defined by

f −α = (√

a −α+√

a −β)(√

a −α+√

a − γ ),

and

g = (√

a −α+√

a −β)(√

a −β +√

a − γ )(√

a − γ +√

a −α),

satisfies

2Q = P.

Furthermore, if {α1, α2, α3} = {α, β, γ }, Di = (αi , 0) ∈ E(k1), i = 1, 2, 3, andQ(i)

= ( f (i), g(i))= Q + Di , then

(2–1) ( f −αi )( f (i) −αi )= (αi −α j )(αi −α j ′),

where { j, j ′} = {1, 2, 3} \ {i}.

3. Local calculations

Given a point P ∈ E(k), let K be the field determined by P as in Lemma 2.1. Forv ∈ Mk , let Kw be the completion of K with respect to a place w lying over v.Then Kw/kv is a Galois extension. Let Iw be the inertia subgroup of Gal(Kw/kv).In this section, we assume that w is nonarchimedean and view it as an valuationfrom Kw onto Z ∪ {∞}.

Lemma 3.1. Suppose E has potential good reduction at a place v of k such thatv(2)= 0. Then for any place w of K lying over v, we have

w(α−β)= w(β − γ )= w(γ −α).

Proof. Suppose on the contrary that

w(γ −α) > w(α−β)= w(β − γ ).

We can find a field extension K of K such that v(α − β) = 2m, m ∈ Z, where vis a place of K lying over w. By our assumption, we have v(β − γ ) = 2m andv(γ −α) > 2m. Consider the elliptic curve E defined by

E : Y 2= X(X − β)(X − γ ),

which was obtained from (1–1) by the change of variables

Y = Y/π3m, X = (X −α)/π2m,

β = (β −α)/π2m, γ = (γ −α)/π2m,


where π is a uniformizer of the prime ideal associated to v in K . Then v(β) = 0and v(γ ) > 0. This implies that E has multiplicative reduction at v. Consequently,v( jE)= v( jE) < 0 which contradicts our hypothesis. �

Now assume that the equation for E is minimal at v. Let Fv be the residue fieldof v and let E be the reduction of E at v. As usual, for P ∈ E(kv), we denote itsimage under the reduction map E(kv)→ E(Fv) by P . Put

E0(kv)= {P ∈ E(kv) | P ∈ Ens(Fv)},

where Ens is the set of nonsingular points of E . We have the following key lemma.Here we retain the notations in Lemma 2.1.

Lemma 3.2. Assume that at v, where v(2) = 0, the Weierstrass equation (1–1)is minimal and E has potential good reduction. For P1, P2 ∈ E(Ov), let Qi =

( fi , gi ) ∈ E(Kw), for i = 1, 2, be such that 2Qi = Pi . If Q1 − Q2 ∈ E0(kv), then

w( f1 −α)= w( f2 −α) and w( f1 −β)= w( f2 −β).

Before we give the proof of Lemma 3.2, we recall some basic facts on the formalgroup associated to an elliptic curve.

Suppose w(α−β)= 2a + ε, where a ∈ N∪{0} and ε = 0 or 1. By Lemma 3.1,w(β − γ )= w(γ −α)= 2a + ε. Consider the change of variables

Y = Y/π3a, X = (X −α)/π2a,

β = (β −α)/π2a, γ = (γ −α)/π2a,

where π is a uniformizer of the prime ideal associated to w. Then

E : Y 2= X(X − β)(X − γ ),

is a minimal Weierstrass equation for E over Kw. For i = 1, 2, let Qi = ( fi , gi ),be the points on E corresponding to Qi . Let E be the formal group associated toE/Kw. For m ≥ 0, set

Em =

{E0(Kw) if m = 0,

E(πmOKw) if m > 0.

Then we have the filtration

· · · ⊂ Em+1 ⊂ Em ⊂ · · · ⊂ E1 ⊂ E0.

Also, recall that we have the exact sequence

0 −→ E1 −→ E0 −→¯Ens −→ 0,

where ¯Ens is the nonsingular part of the reduction of E .


For a point R = (X , Y ) in E(Kw), let t = −X/Y . The following lemma followseasily from [Silverman 1986, Chapter IV].

Lemma 3.3. Let notations be as above.

(1) If m > 0, then

R ∈ Em \ Em+1 ⇐⇒ w(t)= m ⇐⇒(w(X)= −2m and w(Y )= −3m

).

(2) If m = 0 and ε = 0, then

R ∈ E0 \ E1 ⇐⇒ w(t)≤ 0 ⇐⇒(w(X)≥ 0 and w(Y )≥ 0

).

(3) If m = 0 and ε = 1, then

R ∈ E0 \ E1 ⇐⇒ w(t)= 0 ⇐⇒(w(X)= 0 and w(Y )= 0

).

Note that if ε = 0, then E has good reduction at w. In this case, E0 = E(Kw).

Lemma 3.4. Under the hypothesis of Lemma 3.2, suppose that w(α−β)= 2a + ε

and Q = ( f, g) ∈ E0(kv). Then Q ∈ Ea ⊂ E0.

Proof. Recall that the reduction of E is

E : Y 2= (X − α)(X − β)(X − γ ).

The singularity of E is (α, 0).If Q = ( f, g)∈ E0(kv), then w( f −α)≤ 0. Since f = ( f −α)/π2a , g = g/π3a ,

we have w( f )≤ −2a. By Lemma 3.3, we have Q ∈ Ea ⊂ E0. �

Proof of Lemma 3.2. We apply Lemma 2.1 with α1 = α, α2 = β, and α3 = γ . ThenQ′

1 = Q1 + (α, 0), and so on. By (2–1), we have

( f1 −α)( f ′

1 −α)= (α−β)(α− γ ).

This and Lemma 3.1 imply

w( f1 −α)+w( f ′

1 −α)= 2(2a + ε),

and

(3–1) w( f1)+w( f ′

1)= 2ε.

Similarly,

(3–2) w( f2)+w( f ′

2)= 2ε.

First we consider the case where

w( f1 −α)≤ 2a + ε.


Then w( f1) ≤ ε. If w( f1) > 0, then w( f1) = ε = 1. In this situation, E hasadditive reduction at w and (0, 0) is the singularity of the reduction. Therefore,Q1 6∈ E0(Kw). By Lemma 3.4, Q1 − Q2 ∈ Ea ⊂ E0, and consequently Q2 is notin E0(Kw). Hence w( f2) > 0. By (3–1), we also have w( f ′

1) = 1. Repeatingthe above argument, we also conclude that w( f ′

2) > 0. Then (3–2) implies thatw( f2)= w( f ′

2)= 1.Now, assume that w( f1) = −2m ≤ 0. Note that by Lemma 2.1 Qi ∈ E(Ow),

i = 1, 2 and we have w( fi −α)≥ 0. Hence,

(3–3) w( fi )≥ −2a.

This means that Q1 /∈ Ea+1 and Q1 ∈ Em \ Em+1. If a > m, then by Lemma 3.3and Lemma 3.4, we also have

Q2 ∈ Em \ Em+1

and hencew( f2)=−2m. If a = m, then we have Q2 ∈ Ea and hencew( f2)≤−2a.By (3–3), we have w( f2)= −2m, too.

For the case wherew( f1 −α) > 2a + ε,

we consider f ′

1, which, according to (2–1), satisfies

w( f ′

1 −α) < 2a + ε.

Then the argument above can be applied to verify that

w( f ′

2 −α)= w( f ′

1 −α).

We complete the proof by applying (2–1). �

Let K be as given in Lemma 2.1 and let w be a nonarchimedean place of K . Apoint Q = ( f, g) ∈ E(Kw) is called special if

w( f −α) <min{w(α−β),w(β − γ ),w(γ −α)}.

If Q is special, then

w( f −α)= w( f −β)= w( f − γ ).

Put {α1, α2, α3} = {α, β, γ }, and let Q(i) be as in Lemma 2.1.

Lemma 3.5. Suppose that Q(0)= Q ∈ E(Kw) and E has G-type reduction at w

with

w(α1 −α2)= w(α2 −α3)= w(α3 −α1)= ε.


(1) If Q is special and w( f − α1) = ε − e < ε, then for j = 1, 2, 3, Q( j) is notspecial and

w( f ( j)−αi )=

{ε+ e if i = j,

ε if i 6= j.

(2) If every Q( j) is not special for j = 0, 1, 2, 3, then, for every i and j ,

w( f ( j)−αi )= ε.

Proof. Suppose that Q is special. By (2–1),

w( f ( j)−α j )= 2w(α−β)−w( f −α)= ε+ e.

If i 6= j , then

w( f ( j)−αi )= w( f ( j)

−α j +α j −αi )= min(ε+ e, ε)= ε.

If every Q( j), j = 0, 1, 2, 3, is not special, then for every i , w( f ( j)− αi ) ≥ ε.

By (2–1) again, we must have w( f ( j)−αi )≤ ε. �

Lemma 3.6. Suppose that Q ∈ E(Kw) and E has M-type reduction with

ε1 = w(α1 −α2)= w(α1 −α3) < w(α2 −α3)= ε2.

(1) If Q is special and w( f −α1)= ε1 − e < ε1, then, for j = 1, 2, 3, Q( j) is notspecial and

w( f ( j)−αi )=

ε1 + e if i = j = 1,

ε2 + e if i = j = 2, 3,

ε1 if ( j = 1, i 6= 1)or (i = 1, j 6= 1),

ε2 if i, j = 2, 3, j 6= i.

(2) If every Q( j), j = 0, 1, 2, 3, is not special and w( f −α2)= ε1 + e, then

ε1 = w( f −α1)≤ ε+ e = w( f −α3)≤ ε2.

Moreover, for i, j = 1, 2, 3,

w( f ( j)−αi )=

ε1 + e if j = 1, i 6= 1

ε1 if i = 1

ε2 − e if i 6= 1, j 6= 1.

Proof. Most of the proof is similar to that of Lemma 3.5. Only the valuationsof f (1) − αi , i 6= 1, need special calculation. But, since Q(1)

= Q(2)+ D3 and


Q(1)= Q(3)

+ D2, by (2–1), we have

w( f (2) −α2)+w( f (1) −α2)= ε1 + ε2,

w( f (3) −α3)+w( f (1) −α3)= ε1 + ε2. �

4. Unit equations

LetC = {(P, Q) | P ∈ E(OS), 2Q = P}.

For (P1, Q1), (P2, Q2) ∈ C, we define an equivalence relation as follows:

(P1, Q1)∼ (P2, Q2) if and only if Q1 − Q2 ∈ 12E(k).

Let (P1, Q1), . . . , (Pc, Qc) represent all the equivalence classes in C. Then

c ≤ 4 × E(k)/24E(k)≤ 4 × 24r+2.

Now, we fix an equivalence class represented by (Pl, Ql). If (P, Q)∼ (Pl, Ql)

and Q = ( f, g), Ql = ( fl, gl), then the quantities

(4–1)x = ( f −α)/( fl −α), y = ( f −β)/( fl −β),

λ= ( fl −α)/(β −α), µ= (β − fl)/(β −α)

satisfy

(4–2) λx +µy = 1.

Note that Q and Ql determine the same field extension K/k. Let

S = {w | w ∈ MK and w|v, for some v ∈ S′∪ S1 ∪ Sm}.

Using (2–1), we see that x and y are units at every place w not sitting over S ∪

S0 ∪ S1 ∪ Sm . For v ∈ S0, E has additive reduction at v. Therefore,

12E(kv)⊂ E0(kv).

Applying Lemma 3.2 to Q and Ql , we see that (4–2) is an S-unit equation.Now we apply the theory of [Evertse 1984] to bound the cardinality of the equiv-

alence class of (Pl, Ql). We will follow the setting in that paper. Fix a primitivethird root ρ of 1 and put L = K (ρ). Given (P, Q) in the equivalence class of(Pl, Ql), we define x, y, λ, µ by (4–1) and put

ξ = ξ(x, y)= λx − ρµy, η = η(x, y)= λx − ρ2µy, ζ = ζ(x, y)= ξ/η.

We denote by V0 the set of those ζ ∈ L for which an S-unit solution (x, y) of(4–2) exists with λx/µy not a root of one and such that ζ = ζ(x, y). We denoteby V1 the subset consisting of those ζ(x, y) such that x and y are defined by (4–1)


using a point (P, Q) in the equivalence class of (Pl, Ql). We can recover x and yfrom ζ . Therefore, it is enough to bound the number of elements in V1.

Let T be the set of places of L sitting over S and put

A =

( ∏V ∈T

|3|V

)1/2 ∏V ∈T

|λµ|V

( ∏V /∈T

max(|λ|V · |µ|V )

)3

.

Definition 4.1. For V ∈ ML , ζ ∈ L , put

mV (ζ )= mini=0,1,2

(1,max(|1 − ρiζ |V , |1 − ρ−iζ−1|V ).

Lemma 4.2 [Evertse 1984, Lemma 3]. We have∏V ∈T

mV (ζ )≤ 8Ah(ζ )−3 for ζ ∈ V0.

The next lemma follows by direct calculation.

Lemma 4.3. Suppose that V ∈ ML is nonarchimedean and ζ = ζ(x, y) ∈ V0.

(1) If |µy|V < 1, then

mV (ζ ) = |1 − ζ |V = |(1 − ρ)µy|V

< |1 − ρiζ |V , for i 6= 0.

(2) If |λx |V < 1, then

mV (ζ ) = |1 − ρζ |V = |(1 − ρ)λx |V

< |1 − ρiζ |V , for i 6= 1.

(3) If |λx |−1V < 1, then

mV (ζ ) = |1 − ρ2ζ |V = |(1 − ρ)(λx)−1|V

< |1 − ρiζ |V , for i 6= 2.

(4) If |λx |V = |µy|V = 1, then

mV (ζ ) = |1 − ζ |V = |1 − ρζ |V

= |1 − ρ2ζ |V = |1 − ρ|V .

Definition 4.4. For a ζ in V0 and V ∈ T , we choose a ρV ∈ {1, ρ, ρ2} such that

mV (ζ )= min(1,max(|1 − ρV ζ |V , |1 − ρ−1V ζ−1

|V )).

If V is nonarchimedean and we are in case (4) of the preceding lemma, we chooseρV = 1.


For a nonarchimedean place v ∈ S′∪ S1 ∪ Sm , let

Tv = {V ∈ T | V |v}.

Recall that if ζ ∈ V1, there is an associated (P, Q) ∈ C.From now on, we fix the indices so that α1 = α, α2 = β, α3 = γ , Di = (αi , 0),

and as before, we put Q(i)= Q + Di .

Definition 4.5. Let ζ be in V1 and let V be a nonarchimedean place. We say thatζ is of type i , where i = 0, 1, 2, 3, if Q(i) is special at V . If none of the Q(i) isspecial, we say that ζ is of type 4.

Consider the set of numbers∣∣( f ( j)−α j1)/(α j1 −α j2)

∣∣V

and their inverses, where we take j = 0, 1, 2, 3, j1, j2 = 1, 2, 3, and j1 6= j2. Bythe conductor of ζ at V we mean the set CV (ζ ) consisting of all those numbersin this set which are at most one. We list the elements of CV (ζ ) as cV,i withi = 0, 1, 2, . . . and cV,0 = 1. If E has G-type reduction at V , then Lemma 3.5implies that

CV =

{{1, cV,1} if ζ is of type 0, 1, 2, 3;

{1} if ζ is of type 4.

Also, if E has M-type reduction at V , then Lemma 3.6 implies that

CV =

{{1, cV,1, cV,2} if ζ is of type 0, 1, 2, 3;

{1, cV } or {1, cV,1, cV,2} if ζ is of type 4.

Set G = Gal(L/k). Then G acts transitively on Tv and for z ∈ L , σ ∈ G, we have

(4–3) |z|σ(V ) = |σ−1(z)|V .

For z = ( f −α)/(α−β), or z = ( f −β)/(α−β), we have

σ−1(z) ∈ {( f ( j)−αi )/(αi −αi ′) | j = 0, 1, 2, 3, i, i ′

= 1, 2, 3}.

From these facts and Lemma 4.3, we can deduce the next result:

Lemma 4.6. Let v ∈ S′∪ S1 ∪ Sm be a nonarchimedean place and let V0 be a place

in Tv. Then, for a given ζ ∈ V1, the map Tv → {1, ρ, ρ2}, V 7→ ρV , depends only

on the type of ζ at V0. Moreover, if E has G-type reduction at v and CV0 = {1} or{1, cV0,1}, there is a decomposition

Tv = T 0v ∪ T 1

v ,


which depends only on the type of ζ such that

mV =

{1 if V ∈ T 0

v

cV0,1 if V ∈ T 1v .

Also, if E has M-type reduction at v, there is a decomposition

Tv = T 0v ∪ T 1

v ∪ T 2v ,

which depends only on the type of ζ such that

mV =

1 if V ∈ T 0

v ,

cV0,1 if V ∈ T 1v ,

cV0,2 if V ∈ T 2v .

Let v ∈ S′∪ S1 ∪ Sm be a nonarchimedean place. We fix a place V0 in Tv, and

put t iv = #T i

v . If E has G-type reduction at v, define

mv = ct1v

V0,1.

If E has M-type reduction at v, define

mv,1 = ct1v

V0,1 and mv,2 = ct2v

V0,2.

Here we use the convention that if T iv is empty, the associated mv or mv,i is 1.

The following lemma is similar to [Evertse 1984, Lemma 5]. Let S∞ and T∞

be respectively the set of all infinite places in k and L , also, let s∞ = #S∞ andt∞ = #T∞. Note that every place in T∞ is complex, and hence

t∞ = [L : Q]/2 ≤ 4md.

For a real number B with 0< B < 1, put

R(B)= (1 − B)−1 B B/(B−1).

Lemma 4.7. Let B be a real number with 1/2 ≤ B < 1. There exists a set W1 ofcardinality at most

5s′+s1+sm−s∞ × 3t∞ × R(B)s

′+s1+2sm−s∞+t∞−1,

consisting of tuples ((ρV )V ∈T , (0V )V ∈T ) with ρ3V = 1 and 0V ≥ 0 for V ∈ T

and∑

V ∈T 0V = B with the following property: for every ζ ∈ V1 there is a tuple((ρV )V ∈T , (0V )V ∈T ) ∈ W1 such that ζ satisfies

(4–4) min(1, |1 − ρV ζ |V )≤ (8Ah(ζ )−3)0V , for V ∈ T .


Proof. Consider the index set

I = {(w, j) | ( j = 1, w ∈ (S′∪ S1 ∪ T∞) \ (Sm ∪ S∞)) or ( j = 1, 2, w ∈ Sm)}.

Then #I ≤ q := s ′+ s1 + 2sm − s∞ + t∞. For ζ ∈ V1 and (w, j) ∈ I , let

mw, j =

mv if w = v ∈ (S′

∪ S1) \ (Sm ∪ S∞),

mV if w = V ∈ T∞,

mv,1 if w = v ∈ Sm and j = 1,

mv,2 if w = v ∈ Sm and j = 2.

By Lemma 4.2, we have

(4–5)∏

(w, j)∈I

mw, j ≤ 8Ah(ζ )−3, for ζ ∈ V1.

We know form [Evertse 1984, Lemma 4] that there exists a set W of cardinalityat most R(B)q−1 consisting of tuples (8w, j )(w, j)∈I such that for every ζ ∈ V1 thereis a tuple (8w, j )(w, j)∈I such that

mw, j ≤ (8Ah(ζ )−3)8w, j .

Here the tuples can be chosen such that if mw, j = 1, then 8w, j = 0. In particular,if T j

v is empty, we put 8w, j/tjv = 0. We define

0V =

0 if V ∈ T 0

v for some v ∈ S′∪ S1 ∪ Sm \ S∞,

8w,1/t1v if V ∈ T 1

v for some v ∈ (S′∪ S1 ∪ Sm) \ S∞,

8w,2/t2v if V ∈ T 2

v for some v ∈ Sm,

8w, j if V ∈ T∞.

Then inequality (4–4) holds. By Lemma 4.6, there are at most 5s′+s1+sm−s∞ × 3t∞

choices of ρV ’s. �

Now take B = 0.846. The total number of ζ ∈ W1 that satisfy a fixed system(4–4) and for which we have h(ζ )≥ e8/2 is at most 25 (see [Evertse 1984, p. 583]).The cardinality of W1 is at most

5s′+s1+sm−s∞ × 3t∞ × R(B)s

′+s1+2sm−s∞+t∞−1

≤ 5s′+s1+sm−s∞ × 3t∞ × (49/3)s

′+s1+2sm−s∞+t∞−1

≤ 2/25 × (3/49)× (245/3)s′+s1 × (12005/9)sm × (3/245)s∞ × (7)2t∞ .

We note that t∞ is at most 4md . A simple calculation shows that

#W1≤ 2/25 × (3/49)× 72.27(s′

+s1)+3.7sm+8md× (3/245)s∞


By [Evertse 1984, (36)], we have h(λx/µy) ≤ 2h(ζ(x, y)). All of this yields thefollowing lemma.

Lemma 4.8. The total number of (P, Q) ∼ (Pl, Ql) with Q = ( f, g) such thath(( f −α)/( f −β))≥ e8 is at most

6/49 × 72.27(s′+s1)+7.2sm+8md

× (3/245)s∞ .

Proof of Theorem 1.3. We first fix the equivalence class of (Pl, Ql). We followthe argument in [Evertse 1984, p. 583]. Let s = #S. The group of S-units is thedirect product of s multiplicative cyclic groups, one of which is finite. The fraction( f −α)/( f −β) is a S-unit. We assume that for each v ∈ S′

∪ S1 ∪ Sm \ S∞, a placeVv ∈ Tv is chosen. Consider the index set

8 := {(iv)v | iv = 1, 2, 3, 4, 5, v ∈ S′∪ S1 ∪ Sm \ S∞}.

For each φ = (iv)v ∈8, let

V1φ = {ζ ∈ V1

| ζ is of type iv at every v ∈ S′∪ S1 ∪ Sm \ S∞}.

Then by (2–1) and (4–3), under the map

V1→∏

V ∈S\S∞K ∗

V

ζ 7→ (|( f −α)/( f −β)|V )V ,

the image of each V1φ is in a coset of a subgroup which is a direct product of less

than s ′+ s1 + sm − s∞ multiplicative cyclic groups. This shows that, for a fixed φ,

the set of all ( f −α)/( f −β) for which ζ ∈ V1φ is in a coset of a subgroup which

is a direct product of less than s3 := t∞ + s ′+ s1 + sm − s∞ multiplicative cyclic

groups. Let n be a positive integer. Then there is an S-unit z and an element ω ∈ Kbelonging to a fixed set of cardinality at most ns3 which does not depend on f suchthat ( f − α)/( f − β) = ωzn . Let ω be a fixed element of this set and let θ be afixed n’th root of ω. By [Evertse 1984, Lemma 1], the number of nonzero z in Kwith h(θ z) < e8/n is at most 5(2e24/n)[K :Q]. Also, the fraction ( f − α)/( f − β)

determines ζ . Using these and taking n = 49/3, we see that the cardinality of thesubset of V1 consisting of those ζ with h(( f −α)/( f −β)) < e8 is at most

5s′+s1+sm−s∞ × 5ns3(2e24/n)[K :Q]

≤ (245/3)s′+s1+sm−s∞ × 5 × (49/3)t∞ × 8.784md

≤ 5 × 72.27(s′+s1+sm)+10.3md

× (3/245)s∞ .


Therefore,

#C ≤ 4 × |E(k)/24E(k)| × (3/245)s∞ × (6/49 × 72.27(s′+s1)+3.7sm+8md

+ 3/49 × 72.27(s′+s1+sm)+10.3md)

≤ 4 × |E(k)tor/24E(k)tor| × (3/245)s∞ × 24r× 6 × 72.27(s′

+s1)+3.7sm+10.3md

≤ 4 × 6 × |E(k)tor/24E(k)tor| × (3/245)s∞ × 71.64r+2.27(s′+s1)+3.7sm+10.3md .

The map C → E(OS) given by (P, Q) 7→ P is 4 to 1. If s∞ ≥ 2, then

6 × |E(k)tor/24E(k)tor| × (3/245)s∞ ≤ 6 × 242× (3/245)2 < 1,

and the theorem is proved. Otherwise, the number field k has degree at most 2,and the order of the torsion part of the multiplicative group k∗ is at most 6. Inthis case, via Weil pairing, we see that if E(k)tor contains a subgroup of the formZ/NZ × Z/NZ then N ≤ 6. Consequently, we have |E(k)tor/24E(k)tor| ≤ 24 × 6and hence

6 × |E(k)tor/24E(k)tor| × (3/245)s∞ ≤ 36 × 24 × (3/245) < 11,

as we wished to show. �

References

[Chi et al. 2004] W.-C. Chi, K. F. Lai, and K.-S. Tan, “Integral points on elliptic curves over functionfields”, J. Aust. Math. Soc. 77:2 (2004), 197–208. MR 2005g:11093 Zbl 02158910

[Chi et al. ≥ 2006] W.-C. Chi, P.-Y. Huang, and K.-S. Tan, “Uniform bounds for the number ofinteger solutions to Y n

= f (X)”. In preparation.

[Evertse 1984] J.-H. Evertse, “On equations in S-units and the Thue–Mahler equation”, Invent.Math. 75:3 (1984), 561–584. MR 85f:11048 Zbl 0521.10015

[Gross and Silverman 1995] R. Gross and J. Silverman, “S-integer points on elliptic curves”, PacificJ. Math. 167:2 (1995), 263–288. MR 96c:11057 Zbl 0824.11038

[Hindry and Silverman 1988] M. Hindry and J. H. Silverman, “The canonical height and integralpoints on elliptic curves”, Invent. Math. 93:2 (1988), 419–450. MR 89k:11044 Zbl 0657.14018

[Lang 1978] S. Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wis-senschaften 231, Springer, Berlin, 1978. MR 81b:10009 Zbl 0388.10001

[Silverman 1984] J. H. Silverman, “The S-unit equation over function fields”, Math. Proc. Cam-bridge Philos. Soc. 95:1 (1984), 3–4. MR 85e:11018 Zbl 0533.10013

[Silverman 1986] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics106, Springer, New York, 1986. MR 87g:11070 Zbl 0585.14026

[Silverman 1987] J. H. Silverman, “A quantitative version of Siegel’s theorem: integral points onelliptic curves and Catalan curves”, J. Reine Angew. Math. 378 (1987), 60–100. MR 89g:11047Zbl 0608.14021

[Tan 2002] K.-S. Tan, “A 2-division formula for elliptic curves”, National Taiwan University, Janu-ary 2002.



WEN-CHEN CHI


NATIONAL TAIWAN NORMAL UNIVERSITY

TAIPEI

TAIWAN

[email protected]

KING FAI LAI

SCHOOL OF MATHEMATICS AND STATISTICS

UNIVERSITY OF SYDNEY

SYDNEY, NSW 2006AUSTRALIA

[email protected]

KI-SENG TAN


NATIONAL TAIWAN UNIVERSITY

TAIPEI

TAIWAN

[email protected]


ANALYTIC FLOWS ON THE UNIT DISK: ANGULARDERIVATIVES AND BOUNDARY FIXED POINTS

MANUEL D. CONTRERAS AND SANTIAGO DÍAZ-MADRIGAL

We use the concept of angular derivative and the hyperbolic metric in theunit disk D, to study the dynamical aspects of the equilibrium points belong-ing to ∂D of some complex-analytic dynamical systems on D. Our resultsshow a deep connection between the dynamical properties of those equilib-rium points and the geometry of certain simply connected domains of C.As a consequence, and in the context of semigroups of analytic functions,we give some geometric insight to a well-known inequality of Cowen andPommerenke about the angular derivative of an analytic function.

1. Introduction

A (one-parameter) semigroup of analytic functions is any continuous homomor-phism8 : t 7→8(t)= ϕt from the additive semigroup of nonnegative real numbersinto the composition semigroup of all analytic functions which map D into D. Thatis, 8 satisfies the following three conditions:

(a) ϕ0 is the identity in D,

(b) ϕt+s = ϕt ◦ϕs , for all t, s ≥ 0,

(c) ϕt(z) tends to z as t tends to 0, uniformly on compact subsets of D.

It is well-known that condition (c) can be replaced by

(c’) For every z ∈ D, limt→0 ϕt(z)= z.

Semigroups of this type appear in many areas of analysis, such as the theoryof composition operators, the theory of Markov stochastic processes, optimizationtheory and the theory of planar vector fields. In this paper, we are interested in thislast aspect, which we discuss in detail for completeness.

MSC2000: primary 30C20, 30D05, 37F99; secondary 30F45, 37E35.Keywords: angular derivative, fixed points, planar vector fields, semigroups of analytic functions.This research has been partially supported by the Ministerio de Ciencia y Tecnología (projectsBFM2000-1062 and BFM2003-07294-C02-02) and by La Consejería de Educación y Ciencia dela Junta de Andalucía.

253

254 MANUEL D. CONTRERAS AND SANTIAGO DÍAZ-MADRIGAL

Given a semigroup8= (ϕt), it can be proved (see [Shoikhet 2001; Berkson andPorta 1978]) that there exists a unique analytic function G : D → C such that, foreach z ∈ D, the trajectory

γz : [0,+∞)→ R2, t 7→ γz(t) := ϕt(z)

is the solution of the Cauchy problem{x = f (x, y), y = g(x, y),(x(0), y(0))= z,

where f = Re G and g = Im G. Usually, this planar dynamical system is writtenin the form w = G(w), that is, as a complex-analytic dynamical system in D. Wecall G the vector field, and w = G(w) the dynamical system, associated with 8.In operator theory, G is also known as the infinitesimal generator of 8. Thereis a very nice representation, due to Berkson and Porta [1978], of those analyticfunctions on the disk which are generated in this way:

An analytic function G : D → C is the vector field of a semigroup of analyticfunctions8 if and only if there is a point b ∈ D and an analytic function p : D → C

with Re p ≥ 0 such that

G(z)= (b − z)(1 − bz)p(z), z ∈ D.

Moreover, such a representation is unique. The point b is called the Denjoy–Wolffpoint (DW-point) of 8 and p is called the Caratheodory function associated to 8.

Looking at this representation, we see that, whenever b ∈ D, the associateddynamical system has a critical point. However, G never vanishes if b ∈ ∂D.Moreover, it is well known that complex-analytic dynamical systems in simplyconnected domains cannot have limit cycles [Needham and King 1994], so it isnatural to wonder about the “elements” which govern (and how they do it) thedynamics of this system without critical points and without limit cycles. In therest of this section, we will see that the key for understanding this dynamics isthe study of some “critical points” belonging to ∂D. Since, in our setting, G andthe functions (ϕt) are only defined in D, that comment requires a clarification andthis leads us to recall some concepts from complex function theory [Pommerenke1975; 1992; Shapiro 1993].

Let ϕ : D → C be an analytic function and take a ∈ ∂D. We say that L ∈ C∞ isthe angular limit of ϕ in a when z tends to a if, for every α ∈ (0, π/2),

limz∈S(a,α)

z→a

ϕ(z)= L ,

where S(a, α) denotes the Stolz angle with center at a and opening α. The numberL is commonly denoted by 6 limz→a ϕ(z). When 6 limz→a ϕ(z) = a, the point a

ANALYTIC FLOWS ON THE UNIT DISK 255

is called a boundary fixed point of ϕ. If we suppose additionally that ϕ(D) ⊂ D,the angular limit

6 limz→a

ϕ(z)− az − a

always exists and belongs to (0,+∞) ∪ {∞}. This value is denoted by ϕ′(a)and we refer to it as the multiplier of ϕ in a. When ϕ′(a) ∈ (0,+∞), the limit6 limz→a ϕ

′(z) exists and is equal to ϕ′(a). Following the more or less standardusage in iteration theory, we will say that a is attractive if ϕ′(a)∈ (0, 1]; repulsive ifϕ′(a)∈ (1,+∞) and superrepulsive if ϕ′(a)= ∞. These definitions are consistentwith the intuitive geometric meaning of a repulsive or attractive point, thanks tothe celebrated Denjoy–Wolff Theorem [Shapiro 1993, Section 5.1].

Now, we come back to the semigroup 8 = (ϕt) and the associated vector fieldG with DW-point b ∈ ∂D. A point a ∈ ∂D is a boundary fixed point of 8 if ais a boundary fixed point for each ϕt , where t ≥ 0. It is a nontrivial fact due toCowen [1981, Theorem 5.2] that “for each ϕt ” can be replaced by “for some ϕt ”.Moreover, the DW-point b is a boundary fixed point of8 and, indeed, the Denjoy–Wolff Theorem admits the following version for semigroups (see [Berkson andPorta 1978; Siskakis 1985]):

The point b is the unique boundary fixed point of 8 with ϕ′t(b) ≤ 1 for some

(hence for all) t > 0. Moreover, for every z ∈ D, we have limt→+∞ ϕt(z)= b.

Combining the results of [Siskakis 1985] and [Cowen 1981], it is possible todeduce that the dynamical character of the multipliers of all the functions ϕt is thesame. That is, a point ξ ∈ ∂D is attractive (or repulsive, or superrepulsive) for somet if and only if the same happens for all t .

Our study of the system w = G(w) will be carried out by considering a modelflow where the trajectories become straight lines but they fill in a more involvedplanar domain �. In other words, our results will explain the dynamics of w =

G(w) in terms of the geometry of�. This domain� is constructed using the theoryof univalent functions [Berkson and Porta 1978; Heins 1981; Siskakis 1985]: thereis a unique univalent function h : D → C with h(0) = 0 whose image � := h(D)satisfies,

�+ t ⊂� for each t > 0

and such thatϕt(z)= bh−1(h(bz)+ t) for t ≥ 0, z ∈ D.

We call� := h(D)⊂ C the associated planar domain of8. Clearly, if we fix z ∈ D

and consider the trajectory t → γz(t) = ϕt(z) with respect to the vector field G,the corresponding trajectory in the model flow is t → h(z)+ t .

Our dynamical approach to w = G(w) deals with the following topics: typesof trajectories, their ω-limits and α-limits and their relationship with the boundary


fixed points; the multipliers of these points and their dynamic meaning; the slope ofthe trajectories along which they arrive at such a fixed point. We recall that a pointξ ∈ C∞ is called an ω-limit point of a curve 0 : (s1, s2)→ C (−∞≤ s1< s2 ≤+∞)

if there exists a strictly increasing sequence (tn) ⊂ (s1, s2) converging to s2 suchthat 0(tn) → ξ . The set of all ω-limit points of 0 is called the ω-limit set anddenoted by ω(0). The definitions of α-limit and α(0) can be given in a similarway but now the role of s2 is played by the point s1.

From the point of view of ω-limits, and in our context, the result of Berkson andPorta cited above reads as follows:

Let 8 be a semigroup of analytic functions with boundary DW-point b, planardomain � and vector field G. Moreover, let γz be a trajectory of the vector fieldG. Then γz is defined for every t ≥ 0, limt→+∞ γz(t) = b and its largest possibleinterval of definition is (T,+∞) with −∞ ≤ T < 0.

The article contains four sections after this introduction. In Section 2 we statethe main results of the paper. With the aim of making the paper more readable, wehave grouped in Section 3 some facts about the hyperbolic distance on the unit disk.In Section 4 we obtain new results about the relationship among boundary fixedpoints, nontangential convergence and angular derivate. Some of these results mayhave some interest of their own, but here we think of them as necessary ingredientsfor the proof of the main results. In Section 5 we give the proofs of all the resultsstated in Section 2.

2. Statement of the main results

For the analysis of the multipliers of ϕt at the DW-point b, we introduce a geometricquantity associated to the planar domain �. Namely, let us denote by ν(�) thesupremum of all positive numbers β such that there is c ∈ C with{

c + ti : −12β < t < 1

2β}

⊂�.

We point out that this is a well-defined concept since � is open and nonvoid.Clearly, ν(�) ∈ (0,+∞]. It is an exercise to show that, whenever ν(�) < +∞,the number ν(�) coincides with the infimum of all positive β such that, there isc ∈ R with

�⊆{z ∈ C : c −

12β < Im z < c +

12β}.

Bearing in mind the property�+t ⊂�, we see that ν(�) represents the “narrowest”width of an open strip parallel to the real axis and completely enclosing �.

The following theorem explains the relation between ν(�) and the multipliersof ϕt in the DW-point b, when ν(�) <∞.


Theorem 2.1. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain �. The following statements are equivalent.

(1) ν(�) <∞.

(2) For all t > 0, we have ϕ′t(b) < 1.

(3) There is t > 0 such that ϕ′t(b) < 1.

Moreover, when ν(�) <∞, it follows that ϕ′t(b)= exp

(−

π

ν(�)t)

for all t ≥ 0.

From Theorem 2.1 and the fact that ϕ′t(b)≤ 1 for some (and therefore all) t > 0,

we readily deduce:

Corollary 2.2. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain �. The following statements are equivalent.

(1) ν(�)= ∞.

(2) For all t > 0, we have ϕ′t(b)= 1.

(3) There is t > 0 such that ϕ′t(b)= 1.

The relation between boundary fixed points and the backward evolution of thetrajectories is expressed in the next proposition.

Proposition 2.3. Let 8= (ϕt) be a semigroup of analytic functions with boundaryDW-point b, planar domain h(D) = � and vector field G. Moreover, let γz be atrajectory of the vector field G with the maximum possible interval of definition(T,+∞) being −∞ ≤ T < 0.

(1) If −∞< T , the limit

a := lims→T

γz(s) ∈ ∂D,

exists, a is not a boundary fixed point of 8, and h(z)+ T ∈ ∂�.

(2) If T = −∞, the limit

a := lims→−∞

γz(s) ∈ ∂D

exists and a is a boundary fixed point of 8.

In our analysis of backward trajectories, a crucial step will be to develop toolsin order to detect when different trajectories go to the same boundary fixed point.For this, we introduce the planar subset

V (�) := int(⋂

t≥0

(�+ t)).

We will check that V (�)+ t = V (�), for every t ∈ R, so we call it the invariantset associated to �. It is worth mentioning that V (�) can be empty. But, if V (�)


is nonvoid and since it is obviously open, we can write V (�) as the union of thecountable (finite or infinite) family of its different connected components. Thesecomponents will be denoted by V j (�) ( j ∈ J ) and, depending on the case, J = N

or J = {1, . . . , n} for some n ∈ N.It can be proved that every component V j (�) of the invariant set V (�) is an

open strip or a half-plane parallel to the real axis in both cases.

Theorem 2.4. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain �= h(D).

(1) If a component V j (�) of V (�) is a half-plane, then for every z ∈ h−1(V j (�)),the corresponding trajectory γz is defined for all t ∈ (−∞, 0] and the α-limitα(γz)= {b}.

(2) If there is z ∈ D such that the corresponding trajectory γz is defined for allt ∈ (−∞, 0] and α(γz) = {b}, then there exists a component V j (�) of theinvariant set of � which is a half-plane and h(z) ∈ V j (�).

There can be one or two half-open components of V (�) related to b. In eithercase, we have ν(�)= ∞ and, therefore, by Corollary 2.2, ϕ′

t(b)= 1, for all t > 0.Those components of the invariant set which are strips are closely related to the

collection of the repulsive boundary fixed points of 8= (ϕt), that is, those pointsa ∈ ∂D such that ϕ′

t(a) ∈ (1,+∞), for every t > 0.

Theorem 2.5. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain �= h(D).

(1) Let the component V j (�) of V (�) be an open strip. There is a unique repul-sive boundary fixed point ξ(V j (�)) of 8 such that, for every z ∈ h−1(V j (�)),the corresponding trajectory γz is defined for all t ∈ (−∞, 0] and α(γz) =

{ξ(V j (�))}.

(2) The map ξ thus defined between open strip components of the invariant setand repulsive boundary fixed points is bijective.

(3) If there is z ∈ D such that the corresponding trajectory γz is defined for allt ∈ (−∞, 0] and α(γz) is a repulsive boundary fixed point ξ(V j (�)), thenh(z) ∈ V j (�).

Now, we study the multipliers of the functions ϕt at boundary fixed points ξwhich are α-limits of trajectories of the system w = G(w). Clearly, we have threepossibilities: ξ is the DW-point, ξ is repulsive or ξ is superrepulsive. The repulsivecase will be analyzed again by means of the strip components of the invariant set.


Theorem 2.6. Let 8 = (ϕt) be a semigroup of analytic functions with planardomain �. If ξ ∈ ∂D is a repulsive boundary fixed point of 8, then

ϕ′

t(ξ)= exp(

π

β j (�)t), for all t ≥ 0,

where β j (�) is the width (necessarily finite) of the strip V j (�) of the invariant setof � associated to ξ .

These results can be translated to the context of the angular derivative of ananalytic function. Cowen and Pommerenke [1982, Theorem 6.1] showed that ifϕ is an analytic and univalent function in D with ϕ(D) ⊂ D, Denjoy–Wolff pointb ∈ ∂D, ϕ′(b) < 1 and ξ1, ξ2, . . . , ξn are distinct fixed points of ϕ (different fromb), then

n∑j=1

1logϕ′(ξ j )

≤ −1

logϕ′(b)

and, if equality holds, ϕ can be embedded in a semigroup of analytic functions andD\ϕ(D) consists of n−1 analytic arcs. Indeed, some geometric motivation of thisinequality can be read in that paper.

Now, if we suppose that ϕ can be embedded in a semigroup of analytic functions,by Theorems 2.1 and 2.6, we can guarantee that, for some t > 0,

ϕ′(b)= exp(

−π

ν(�)t)

and ϕ′(ξ j )= exp(

π

β j (�)t).

Therefore, we can always rewrite the inequality above as

n∑j=1

β j (�)≤ ν(�).

This inequality reflects the geometric fact that the narrowest strip including�mustcontain the family of the disjoint strips V j (�).

Finally, we treat the superrepulsive case.

Theorem 2.7. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain �= h(D) as in the introduction. If there is a pointz ∈ D such that

h(z) ∈

(⋂t≥0

(�+ t))

\

(⋃j∈J

V j (�)

)and the corresponding trajectory γz is defined for all t ∈ (−∞, 0] and α(γz)= {ξ},then ξ is a superrepulsive boundary fixed point of8 and γz is the unique trajectoryof the planar dynamical system with α(γz)= {ξ}.


This does not necessarily account for all superrepulsive fixed points: there aresemigroups with a superrepulsive point ξ where there is no trajectory γz withα(γz)= {ξ}. One example is the following: Consider the planar subset

� :={z ∈ C : −

12 < Im z < exp(Re z)− 1

2

},

and take a Riemann map h from D onto � with h(0)= 0. Now, the semigroup weare looking for is

ϕt(z) := h−1(h(z)+ t), z ∈ D.

The inequality of Cowen and Pommerenke mentioned above implies that thenumber of repulsive fixed points on the boundary of the unit disk is denumerable.However, the number of boundary fixed points can be uncountable (of course,most of them must be superrepulsive). To build such an example, consider the setof rational numbers of the interval (−1, 1), say {α(n) :n ∈ N}, and the planar subset

� := {z ∈ C : −1< Im z < 1} \

(⋃n∈N

((−∞,−n] +α(n)i))

and take a Riemann map h from D onto � with h(0) = 0. Now, by Theorem 2.7,the semigroup of analytic functions given by

ϕt(z) := h−1(h(z)+ t), z ∈ D

has a nondenumerable set of boundary fixed points. This example was suggested tothe authors by Ricardo Perez-Marco. We thank him for this example and other in-teresting remarks concerning this paper. Additional information about the amountof fixed points of analytic self-maps on D can be read in [Cowen and Pommerenke1982, Section 2].

Our next step is the analysis of the slopes of the trajectories of the system.As before, this requires recalling some notation first and distinguishing betweenseveral cases. These cases are related to the evolution of the distance of a fixedpoint to the boundary of the domain � and it suggests to introduce the followingnotation:

δ�(w)= inf {|w− z| : z ∈ ∂�} , w ∈�.

Given a curve 0 : t ∈ (s1, s2)→0(t)∈ D with a singleton ω-limit ω (0)= {ζ } ⊂

∂D, the set of the slopes of 0 when arriving at ζ will be denoted by Slope+(0)

and it is defined as the ω-limit of the curve

t ∈ (s1, s2)→ Arg(1 − ζ 0(t)) ∈(−π2 ,

π2

),

where Arg(z) denotes the principal argument of z. It is well-known that Slope+(0)

is a nonempty, compact, and connected subset of[−π2 ,

π2

]and we talk about the


slope of 0 at ζ if Slope+(0) is a single point. Something similar can be done withα-limits and the definition of Slope−(0) is self-explanatory.

In view of the continuous version of the Denjoy–Wolff Theorem, we know thatlimt→+∞ γz(t) → b, so we may ask about the slope of the forward trajectory γz

when arriving at b. Our next theorem answers precisely this question when ν(�)is finite.

Theorem 2.8. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain �. If ν(�) < ∞, then, for each z ∈ D, the setSlope+(γz) is a single point and it belongs to

(−π2 ,

π2

). That is, the (forward)

trajectory γz tends to b with a fixed slope and never tangentially. Moreover, givenα ∈

(−π2 ,

π2

), there is z ∈ D such that Slope+(γz)= {α}.

The next theorem treats the case ν(�)= +∞. We notice that, given w ∈�, thelimit lims→+∞ δ�(w+ s) always exists and belongs to (0,+∞].

Theorem 2.9. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b, planar domain � and ν(�)= +∞.

(1) If there is w ∈ � such that lims→+∞ δ�(w + s) = +∞, then all the setsSlope+(γz) are identical, when z runs the whole D.

(2) If there is w ∈� such that lims→+∞ δ�(w+s)∈ (0,+∞), then all the trajec-tories γz (z ∈ D) tends tangentially to b. That is, Slope+(γz) is a single pointand it is equal to −

π2 or π

2 .

We conjecture that, in case (1) of this theorem, all the subsets Slope+(γz) areindeed singletons, so it would be possible to speak about the common slope of thetrajectories of the system.

We now study the slopes of backward trajectories. The analysis of the setsSlope−(γz) will be done first, when the α-limit of γz is the DW-point.

Theorem 2.10. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain �.

(1) If the α-limit of a trajectory γz is b, then the set Slope−(γz) is a single pointand it is equal to −

π2 or π

2 . That is, the backward trajectory γz tends tangen-tially to b.

(2) Given two trajectories γz1 and γz2 where b is their α-limit and such that h(z1)

and h(z2) belong to the closure of the same half-plane component of V (�),then Slope−(γz1)= Slope−(γz2).

Finally, we study the slopes of the backward trajectories reaching a repulsiveboundary fixed point.

Theorem 2.11. Let 8 = (ϕt) be a semigroup of analytic functions with boundaryDW-point b and planar domain � and ξ(V j (�)) a repulsive boundary fixed point.


(1) For every z ∈ h−1(V j (�)), the set Slope−(γz) is a single point and it belongsto(−π2 ,

π2

). That is, the backward trajectory γz tends to ξ(V j (�)) with a

fixed slope and never tangentially. Moreover, given α ∈(−π2 ,

π2

), there is

z ∈ h−1(V j (�)) such that Slope−(γz)= {α}.

(2) If there is z ∈ D such that the trajectory γz is defined for all t ∈ (−∞, 0],α(γz) = {ξ(V j (�))} and h(z) ∈ V j (�) \ V j (�), then Slope−(γz) is a singlepoint and it is equal to −

π2 or π

2 . That is, the backward trajectory γz tendstangentially to ξ(V j (�)).

3. The hyperbolic metric

We recall here some facts and notations about the hyperbolic metric; for detailedexposition and proofs, see [Shapiro 1993; Milnor 1999].

Definition 3.1. Given p and q two points of D and γ : [a, b] → D a piecewise C1

function with γ (a)= p and γ (b)= q, we define the hyperbolic length of γ as

lD(γ )= 2∫ b

a

|γ ′(t)| dt1 − |γ (t)|2

.

The hyperbolic distance or Poincare distance from p to q is the length of theshortest curve from p to q , that is,

ρD(p, q)= infγ

lD(γ )

where γ runs through all piecewise C1 curves from p to q.

It is not difficult to see that the hyperbolic distance is an unbounded completemetric on D and that it induces its usual Euclidean topology. Moreover, there is acurve in D where the infimum that appears in the definition is attained.

Proposition 3.2. Let p and q be two points of the unit disk D. Then the followingassertions are true.

(1) The hyperbolic distance can be calculated using the pseudo-hyperbolic dis-tance, namely

ρD(p, q)= log1 + d(p, q)1 − d(p, q)

where d(p, q) is the pseudo-hyperbolic distance between p and q given by

d(p, q)=

∣∣∣∣ p − q1 − pq

∣∣∣∣.(2) If ϕ is an automorphism of the unit disc, then

ρD(ϕ(p), ϕ(q))= ρD(p, q).


(3) Every holomorphic self-map ϕ of D is a contraction in the hyperbolic metric,that is,

ρD(ϕ(p), ϕ(q))≤ ρD(p, q).

The key points in the study of angular derivative using the hyperbolic distanceare the following two results. The first one is a slight generalization of [Shapiro1993, p. 159, equality (8)].

Proposition 3.3. Let ϕ be a holomorphic self-map of D. If ξ ∈ ∂D is a fixed pointof ϕ with ϕ′(ξ) 6= ∞ and (zn) is a sequence in D such that limn Arg(1 − ξ zn) =

θ ∈(−π2 ,

π2

), then

limn→∞

ρD (zn, ϕ (zn))= log

∣∣ϕ′(ξ)+ e−2iθ∣∣+ ∣∣ϕ′(ξ)− 1

∣∣∣∣ϕ′(ξ)+ e−2iθ∣∣− |ϕ′(ξ)− 1|

.

In particular, limn→∞ ρD (zn, ϕ (zn))≥∣∣log

(ϕ′(ξ)

)∣∣ and if limn Arg(1− ξ zn)= 0,then

limn→∞

ρD (zn, ϕ(zn))=∣∣logϕ′(ξ)

∣∣ .Proposition 3.4. Let ϕ be a holomorphic self-map of D. If b ∈ ∂D is the Denjoy–Wolff point of ϕ, and z, z0 two points of D such that the sequence (zn), given byzn+1 = ϕ(zn) for all n, converges to b nontangentially, then

limn→∞

ρD(z, zn)

n=∣∣logϕ′(b)

∣∣ .Proof. We know that

limn→∞

ϕ(zn)= b and limn→∞

ϕ(zn)− bzn − b

= ϕ′(b) ∈ (0, 1].

So,

limn→∞

|zn+1 − b|

|zn − b|= lim

n→∞

∣∣∣∣ϕ (zn)− bzn − b

∣∣∣∣= ϕ′(b).

Therefore,lim

n→∞|zn − b|

1/n= ϕ′(b).

Since (zn) converges nontangentially to b, there is k > 1 such that |zn − b| ≤

k(1 − |zn|) for all n ∈ N. Hence

|zn − b|1/n

≤ k1/n(1 − |zn|)1/n

≤ k1/n(|zn − b|)1/n

for all n. Now, we have limn→∞(1 − |zn|)1/n

= ϕ′(b). Moreover

limn→∞

(1 + |zn|

1 − |zn|

)1/n

=1

ϕ′(b)∈ [1,∞)


and we obtain

limn→∞

ρD(0, zn)

n= lim

n→∞

ρD(0, |zn|)

n= lim

n→∞

1n

log1 + |zn|

1 − |zn|

= log1

ϕ′(b)=∣∣logϕ′(b)

∣∣.Finally, bearing in mind that

ρD(0, zn)− ρD(0, z)n

≤ρD(z, zn)

n≤ρD(0, z)+ ρD(0, zn)

n,

we obtain

limn→∞

ρD(z, zn)

n= lim

n→∞

ρD(0, zn)

n=∣∣logϕ′(b)

∣∣ . �

Remark. A result similar to the proposition can be given for backward iterationsequences for ϕ that converge to a fixed point.

Let h be an univalent function of the unit disk D onto a simply connected domain� C. We shall use the function h to transfer the notion of hyperbolic distancefrom D to �. More precisely, we define the hyperbolic distance on � by

ρ�(h(p), h(q)) := ρD(p, q)

for all p, q ∈ D. Moreover, given 0 a piecewise C1 curve in �, the hyperboliclength of 0 is given by l�(0) := lD(h−1(0)). Thus, the hyperbolic metric in � isinvariant under the action of conformal automorphisms of � and induces the usualEuclidean topology.

A first approach to the relation between the geometry of the domain � and thehyperbolic distance involves the inequalities

12

∫0

|dw|

δ�(w)≤ l�(0)≤ 2

∫0

|dw|

δ�(w).

The first inequality leads to an estimate of how the hyperbolic distance increase asone moves toward the boundary.

Proposition 3.5 (Distance Lemma [Shapiro 1993, p. 157]). If � C is a simplyconnected domain, then for P, Q ∈�, we have

ρ�(P, Q)≥12

log(

1 +|P − Q|

min{δ�(P), δ�(Q)}

).

If we have two simply connected domains�1 and�2 such that�1 ⊂�2(C, wehave a relation between their corresponding hyperbolic metrics. Take h1 a Riemann


map of�1 and h2 a Riemann map of�2. Then ϕ= h−12 ◦h1 is an analytic self-map

of D. So, it is a contraction in the hyperbolic metric, that is,

ρD(ϕ(p), ϕ(q))≤ ρD(p, q)

for all p, q ∈ D. In particular, given P, Q ∈�1, we have

ρ�2(P, Q)= ρD

(h−1

2 (P), h−12 (Q)

)= ρD

(ϕ ◦ h−1

1 (P), ϕ ◦ h−11 (Q)

)≤ ρD

(h−1

1 (P), h−11 (Q)

)= ρ�1(P, Q).

So we have the following well-known result.

Proposition 3.6. Given two simply connected domains �1 and �2 such that �1 ⊂

�2 ( C and P, Q ∈�1, we have

ρ�2(P, Q)≤ ρ�1(P, Q).

4. Fixed points and nontangential convergence

In this section we present a first approach to the relationship between fixed pointsof the semigroup (ϕt) and the geometry of the domain �. Roughly speaking, eachhorizontal half-line in � induces a curve in the unit disc whose end point is afixed point of the functions of the semigroup, and the distance from the pointsof the half-line to the boundary of � determines whether or not convergence istangential.

The starting point is the following well-known result: if 0 : [0,∞) → � isany curve with lims→∞ 0(s) = ∞ (and h is an univalent function on D such that� = h(D)), then there exists w ∈ ∂D such that lims→∞ h−1(0(s)) = w [Shapiro1993, p. 162]. Our first result is a necessary condition to guarantee that, given twodifferent curves in �, the corresponding curves in the unit disk have the same endpoint.

Lemma 4.1. Suppose h is a univalent function on D and � = h(D) and that0i : [0,∞)→�, i = 1, 2, are two Jordan arcs with 01(s) 6=02(s ′) for all (s, s ′) 6=

(0, 0), 01(0)=02(0), lims→∞ 01(s)= lims→∞ 02(s)=∞, and that one of the twoconnected components of the complement of 01[0,∞)∪02[0,∞) in C is includedin �. Then lims→∞ h−1(01(s))= lims→∞ h−1(02(s)).

Proof. Denote by 2 the connected component of the complement of 01[0,∞)∪

02[0,∞) that is included in � and

ω(2) :={b ∈ D : there is {wn} ⊂2, with |wn| ↗ ∞ and h−1(wn)→ b

}.

Notice that ∂2 is a Jordan curve in the Riemann sphere whose boundary is

01[0,∞)∪02[0,∞)∪ {∞}.


Since h is continuous on D, we have ω(2)⊆ ∂D. Moreover, lims→∞ h−1(01(s))and lims→∞ h−1(02(s)) are in ω(2). So the proof is finished if we obtain thatω(2) is a single point. First of all, notice that ω(2) is closed.

Now we focus our attention on checking that ω(2) is connected. Suppose thatthere are two closed, disjoint, and nonempty subsets A and B of ω(2) such thatω(2) = A ∪ B. Let K = d(A, B) > 0. For each natural number n, we can takean ∈ A, bn ∈ B, and wn, wn ∈2, such that∣∣an − h−1(wn)

∣∣< K/3,∣∣bn − h−1(wn)

∣∣< K/3,

and |wn|> |wn|>max {|wn−1|, n}. In particular,

d(h−1(wn), A) < K/3 and d(h−1(wn), A) > 2K/3 for all n.

Claim. Given wn and wn in 2 with |wn| ↗ ∞ and |wn| ↗ ∞, there is a curve γn

in 2, for each n, from wn to wn such that minw∈γn |w| goes to ∞ as n goes to ∞.(The claim will be proved after the proof of the lemma is complete.)

By the continuity of the function h and taking γn the curve given by the claim,there is zn ∈ γn such that d(h−1(zn), A) = K/2 and |zn| → ∞. This is a contra-diction since any point of accumulation of the sequence (h−1(zn)) is in ω(2) butit is neither in A nor in B.

Summing up, ω(2) is nonempty, connected, and compact. Suppose that ω(2)is not a single point. On the one hand, recall that h has radial limits a.e. on ∂D

[Shapiro 1993, p. 162]. On the other hand, using Lehto–Virtanen Theorem weobtain that ω(2) ∂D. So, take a nontrivial subarc ϒ of ω(2) such that if b ∈ϒ ,then −b /∈ ω(2). Take wn and wn in 2 with |wn| ↗ ∞ and |wn| ↗ ∞ suchthat h−1(wn)→ b1 and h−1(wn)→ b2, where b1 and b2 are the extreme points ofω(2). By the claim, for each n, there is a curve γn in 2, from wn to wn, such thatminw∈γn |w| goes to ∞ as n goes to ∞. So, if b ∈ϒ is different from the extremepoints of ω(2) and h has radial limit in b, there is rn → 1 such that h(rnb) ∈ γn .Hence h(rnb)→ ∞. Therefore, we have that limr→1 h(rb)= ∞. That is, h has aradial limit equal to ∞ a.e. on ϒ . But this contradicts that this radial limit is nota.e. constant on any subarc of ∂D [Shapiro 1993, p. 162]. �

Proof of the claim. Since2 is a Jordan domain in the Riemann sphere, its Riemannmapping µ has a bijective and bicontinuous extension to the closed unit disc [Mil-nor 1999, Theorem 17.16]. Denote by γn the segment [µ−1(wn), µ

−1(wn)] ⊂ D.

The curve we are looking for is γn = µ(γn). In effect, take zn ∈ γn such that|zn| = minw∈γn |w|. Then there is λn ∈ [0, 1] such that µ−1(zn) = λnµ

−1(wn)+

(1 − λn)µ−1(wn). Since (wn) and (wn) converge to ∞, we see that (µ−1(zn))

converges to µ−1(∞). Therefore, (zn) goes to ∞ as n tends to ∞. �


Now, it is time to introduce the relationship between half-lines in � and fixedpoints of the functions ϕt .

Lemma 4.2. Fix t > 0 and a ∈ �. Take κ ∈ {−1, 1}. Suppose that the curve0(s) = a + κst , with s ≥ 0, is in �. Then there is a point ξ ∈ ∂D such thatlims→∞ h−1(a + κst)= ξ and ξ is a fixed point of ϕt .

Proof. We know that there is a point ξ ∈ ∂D such that lims→∞ h−1(a + κst) = ξ .Notice that

lims→∞

ϕt(h−1(a + κst))= lims→∞

h−1(a + κst + t)= lims→∞

h−1(a + κ(s + κ)t)

= lims→∞

h−1(a + κst)= ξ.

Therefore, by Lindelof’s Theorem, ξ is a fixed point of ϕt and 6 limz→ξ ϕt(z)= ξ .�

To calculate the derivative at a fixed point we use Propositions 3.3 and 3.4. Wehave to estimate ρD

(z1, ϕt(z2)

)as ϕt(z2) converges to ξ nontangentially. If we take

zi = h−1(a + κsi t), we have

ρD

(h−1(a + κs1t), ϕt(h−1(a + κs2t))

)= ρ�(a + κs1t, a + κs2t + t).

The way to calculate this hyperbolic distance depends on the type (repulsive orattractive) of fixed point we have. But first we have to check that h−1(a + κst)converges nontangentially to ξ . On the one hand, when κ = 1 and the derivativeϕ′

t(ξ) is less than 1, this will be done using the following result:

Lemma 4.3 [Cowen 1981, Lemma 2.1]. Suppose ϕ be a holomorphic self-map ofD, and has Denjoy–Wolff point b ∈ ∂D with ϕ′(b) < 1. Then, for any z in D, thesequence (ϕn(z)) converges nontangentially to b.

On the other hand, nontangential convergence will be characterized in terms ofthe euclidean distance from the point a + κst to the boundary of �. First we needthe following lemma that, roughly speaking, says that to each fixed point with finitederivative corresponds a “tube” in �.

Lemma 4.4. Let ϕ be a holomorphic self-map of D, with ϕ(z)= h−1(h(z)+ λ), hunivalent, λ > 0, � = h(D), and ξ ∈ ∂D a fixed point of ϕ with ϕ′(ξ) 6= ∞. Thenthere is a positive number ε = ε(ξ) such that δ�(h(rξ))≥ ε for all r ∈ [0, 1).

Proof. To simplify the notation, we set

d = ϕ′(ξ)= limr→1−

ξ −ϕ(rξ)ξ − rξ

∈ (0,∞).


Thus there is r0 < 1, such that r0 < r < 1 implies |ξ−ϕ(rξ)| ≤ (d +1)(1−r). So,we have

|rξ −ϕ(rξ)| ≤ |ξ −ϕ(rξ)| + |ξ − rξ | ≤ (d + 2)(1 − r),

whenever r > r0. Hence, estimating the hyperbolic distance by integrating alongthe segment [rξ, ϕ(rξ)], we get

ρD(rξ, ϕ(rξ))≤ 2∫

[rξ,ϕ(rξ)]

ds(x)δD(x)

.

We have to distinguish two cases. On the one hand, if |ϕ(rξ)| ≤ r , we have

ρD(rξ, ϕ(rξ))≤ 2∫

[rξ,ϕ(rξ)]

ds(x)δD(x)

≤ 2|rξ −ϕ(rξ)|

1 − r≤ 2(d + 2).

On the other hand, if |ϕ(rξ)|> r , we have

ρD

(rξ, ϕ(rξ)

)≤ 2

∫[rξ,ϕ(rξ)]

ds(x)δD(x)

≤ 2|rξ −ϕ(rξ)|1 − |ϕ(rξ)|

≤ 2(d + 2)1 − r

1 − |ϕ(rξ)|,

where the second inequality follows from Julia’s Lemma [Cowen and MacCluer1995, p. 49]; from this we then get

ρD

(rξ, ϕ(rξ)

)≤ 2(d + 2)d

1 + |ϕ(rξ)|1 + r

(1 − r)2

|ξ −ϕ(rξ)|2≤ 4(d + 2)d

∣∣∣∣ ξ − rξξ −ϕ(rξ)

∣∣∣∣2.Since

limr→1−

ξ − rξξ −ϕ(rξ)

=1d,

we see that

limr→1

4(d + 2)d∣∣∣∣ ξ − rξξ −ϕ(rξ)

∣∣∣∣2 = 4d + 2

d.

Hence, there is r0 ≤ r1 < 1 such that ρD

(rξ, ϕ(rξ)

)is bounded for r ∈ (r1, 1).

So far, we have found that there is a constant M > 0 such that

ρ�(h(rξ), h(rξ)+ λ

)= ρD

(rξ, ϕ(rξ)

)≤ M for all r ∈ [0, 1).

Now, by the Distance Lemma (Proposition 3.5),

2M ≥ log(

1 +|h(rξ)− h(rξ)− λ|

min {δ�(h(rξ)), δ�(h(rξ)+ λ)}

)≥ log

(1 +

λ

δ�(h(rξ))

).

That is, δ�(h(rξ))≥ λ/(e2M− 1) > 0 for all r . �

It is worth mentioning that this lemma implies that if ξ ∈ ∂D is a fixed point ofϕ with ϕ′(ξ) 6= ∞, then limr→1− h(rξ)= ∞.


Now we can give a sufficient condition to get the nontangential convergence.Since ϕt(rξ) converges nontangentially (as r → 1) to the fixed point ξ , we willcompare the curve r 7→ ϕt(rξ) with the curve s 7→ h−1(a + κst).

Proposition 4.5. Let (ϕt) be a semigroup of analytic functions with planar domain�= h(D). Let ξ ∈ ∂D be a nonsuperrepulsive (ϕ′

t(ξ) 6= ∞ for all t > 0) fixed pointof (ϕt). Take κ ∈ {−1, 1}. Suppose that there is s0 such that

(1) there is α > 0 such that δ�(a + κst)≥ α for all s ≥ s0;

(2) there is β <∞ such that for each s ≥ s0, there is r(s) such that Re h(r(s)ξ)=a + κst ,

[a + κts, h(r(s)ξ)+ t

]⊂�, and |h(r(s)ξ)− (a + κst)| ≤ β.

Then h−1(a + κst) converges nontangentially to ξ as s goes to ∞.

Proof. Bearing in mind [Shapiro 1993, p. 171, Exercise 4], it is enough to find aconstant M such that ρD

(h−1(a + κts), γ

)≤ M for all s large enough where γ is

the segment (−ξ, ξ). Fix s > 0. We have

ρD

(h−1(a + κts), γ

)≤ inf

0≤r<1ρD

(h−1(a + κts), rξ

)≤ inf

0≤r<1

(ρD(rξ, ϕt(rξ))+ ρD(h

−1(a + κts), ϕt(rξ))).

Since ϕt(rξ) converges nontangentially (as r → 1) to the fixed point ξ , there isa constant m such that ρD(rξ, ϕt(rξ))≤ m. So we just have to control

inf0≤r<1

ρD

(h−1(a + κts), ϕt(rξ)

)as s tends to ∞. Notice that

inf0≤r<1

ρD

(h−1(a + κts), ϕt(rξ)

)= inf

0≤r<1ρ�(a + κts, h(rξ)+ t

)≤ 2 inf

0≤r<1inf0

∫0

ds(x)δ�(x)

,

where the last infimum is taken over all curves that goes from a +κts to h(rξ)+ tin �. Take r(s) such that Re h(r(s)ξ)+ t = κst . Now we estimate the hyperbolicdistance by integrating along the segment [a + κts, h(r(s)ξ)+ t]:

inf0≤r<1

ρD(h−1(a + κts), ϕt(rξ))

≤ 2 inf0≤r<1

inf0

∫0

ds(x)δ�(x)

≤ 2∫

[a+κts,h(r(s)ξ)+t]

ds(x)δ�(x)

≤ 2|a + κts − h(r(s)ξ)− t |

min{δ�(a + κts), δ�(h(r(s)ξ)+ t)}≤ 2

|a + κts − h(r(s)ξ)− t |min{δ�(a + κts), δ�(h(r(s)ξ))}

.


Now, by Lemma 4.4, there is ε = ε(ξ) such that δ�(h(rξ)) ≥ ε for all r ∈ [0, 1).Therefore

ρD

(h−1(a + κts), γ

)≤ 2

β + tmin

{α, ε

}for s ≥ s0. Therefore, ρD

(h−1(a + κts), γ

)is bounded as s goes to ∞. �

Lemma 4.6. Let ϕ be a holomorphic self-map of D, with ϕ(z)= h−1(h(z)+ λ), hunivalent, λ > 0 and �= h(D) with Denjoy–Wolff point equal to b ∈ ∂D and suchthat ϕ′(b) = 1. If (pn) is a sequence in D which converges nontangentially to b,then

limn→∞

δ�(h(pn))= ∞.

Proof. Take a subsequence (pnk ) of (pn) such that limk Arg(1 − b pnk ) = θ . Thenθ ∈

(−π2 ,

π2

)and, by Proposition 3.3,

limn→∞

ρ�(h(pnk ), h(pnk )+ λ

)= lim

n→∞ρD

(pnk , ϕ(pnk )

)= log

∣∣ϕ′(b)+ e−2iθ∣∣+ ∣∣ϕ′(b)− 1

∣∣∣∣ϕ′(b)+ e−2iθ∣∣− |ϕ′(b)− 1|

= log 1 = 0.

Because the choice of a subsequence was arbitrary we can deduce from this thatlimn→∞ ρ�

(h(pn), h(pn)+λ

)=0. Moreover, by the Distance Lemma (Proposition

3.5), we have

ρ�(h(pn), h(pn)+ λ

)≥

12

log(

1 +|h(pn)− h(pn)− λ|

min {δ�(h(pn)), δ�(h(pn)+ λ)}

)≥

12

log(

1 +λ

δ�(h(pn))

).

Hence, limn→∞ δ�(h(pn))= ∞. �

5. Proofs of the main results

Proof of Theorem 2.1. Siskakis [1985, Theorem 1.7] proved that there is a number0 < r ≤ 1 such that ϕ′

t(b) = r t for all t > 0. Therefore, it is clear that (2) isequivalent to (3).

(1)⇒ (2). By assumption, there is a real number a such that

�⊆2 :={z ∈ C : a −

12ν(�) < Im z < a +

12ν(�)

}.

A Riemann map of 2 is given by 8(z) = (ν(�)/π) Log((1 − z)/(1 + z))+ ai .Fix t > 0. Take c ∈ R such that ai + c ∈�. Consider the points wn = ai + c + nt .Notice that wn ∈�. We will apply Proposition 3.4 with zn = h−1(wn) and ϕ = ϕt .So we have to check that (zn) converges nontangentially to b. To obtain this wewill apply Proposition 4.5 with ξ = 1 and κ = 1. Recall that 0 < ϕ′

t(b) ≤ 1. On


the one hand, in our case, the function s 7→ δ�(ai + c + st) is nondecreasing. So,there is α > 0 such that δ�(ai +c+ st)≥ α for all s ≥ 0. On the other hand, h(rb)converges to ∞ and, bearing in mind that b is attractive and that�⊆2, we see thatRe h(rb) converges to +∞ as r tends to 1. So, the assumption (2) of Proposition4.5 is satisfied for s large enough. Hence we can apply this proposition and obtainthat (zn) converges nontangentially to b. Now we are ready to apply Proposition3.4:

− logϕ′

t(b)= limn→∞

ρD

(h−1(ai + c), ϕt(h−1(wn))

)n

= limn→∞

ρ�(ai + c, ai + c + nt + t)n

.

By Proposition 3.6, we have

limn→∞

ρ�(ai + c, ai + c + nt + t)n

≥ limn→∞

ρ2(ai + c, ai + c + nt + t)n

.

Therefore

− logϕ′

t(b)≥ limn→∞

ρ2(ai + c, ai + c + nt + t)n

= limn→∞

ρD

(8−1(ai + c),8−1(ai + c + nt + t)

)n

= limn→∞

1nρD

(1 − exp(cπ/ν(�))1 + exp(cπ/ν(�))

,1 − exp((c + nt + t)π/ν(�))1 + exp((c + nt + t)π/ν(�))

)

= limn→∞

1n

log

1 +

∣∣∣∣1 − exp((nt + t)π/ν(�))1 + exp((nt + t)π/ν(�))

∣∣∣∣1 −

∣∣∣∣1 − exp((nt + t)π/ν(�))1 + exp((nt + t)π/ν(�))

∣∣∣∣

= limn→∞

1n

log exp( π tν(�)

(n + 1))

=π

ν(�)t.

That is, ϕ′t(b)≤ exp(−π t/ν(�)) < 1 for all t > 0.

(2) ⇒ (1) Fix t > 0 and take an arbitrary λ < ν(�). We will prove shortly thatϕ′

t(b)≥ exp(−π t/λ), that is, λ≤ π t/(− logϕ′t(b)). Therefore,

ν(�)≤π t

− logϕ′t(b)

<∞.

Moreover, once we have obtained this inequality and bearing in mind the proof ofthe implication (1) ⇒ (2), we will have shown that ν(�) < ∞ implies ϕ′

t(b) =

exp(−π t/ν(�)).To obtain that ϕ′

t(b)≥exp(−π t/λ), we again apply Proposition 3.4. Take a pointa such that

[a −

12λi, a +

12λi

]⊂ �. Bearing in mind the geometric properties of


�, we have

2= {z ∈ C : Re z > Re a and Im a −12λ < Im z < Im a +

12λ} ⊆�.

Recall that a Riemann map of 2 is given by

9(z)= −iλ

πsin−1

(i1 − z1 + z

)+ a.

In fact,

9−1(w)=1 + i sin(iπ(w− a)/λ)1 − i sin(iπ(w− a)/λ)

.

Consider the points wn = a + nt ∈ 2 ⊂ �. We will apply Proposition 3.4 withzn = h−1(wn) and ϕ = ϕt . First of all, since ϕ′

t(b) < 1, Lemma 4.3 says that (zn)

converges nontangentially to b. Hence

− logϕ′

t(b)= limn→∞

ρD

(h−1(a + t), ϕt(h−1(a +nt))

)n

= limn→∞

ρ�(a + t, a +nt + t)n

.

By Proposition 3.6, ρ�(a + t, a + nt + t)≤ ρ2(a + t, a + nt + t). Therefore,

− logϕ′

t(b)≤ limn→∞

ρ2(a + t, a + (n + 1)t

)n

= limn→∞

ρD

(9−1(a + t),9−1(a + (n + 1)t)

)n

= limn→∞

1nρD

(1 + i sin(iπ t/λ)1 − i sin(iπ t/λ)

,1 + i sin(iπ t (n + 1)/λ)1 − i sin(iπ t (n + 1)/λ)

)= lim

n→∞

1nρD

(2 + e−π t/λ

− eπ t/λ

2 − e−π t/λ + eπ t/λ ,2 + e−π t (n+1)/λ

− eπ t (n+1)/λ

2 − e−π t (n+1)/λ + eπ t (n+1)/λ

)= lim

n→∞

1n

log(

eπ t (n+1)/λ− e−π t (n+1)/λ

eπ t/λ − e−π t/λ

)=π tλ.

That is, ϕ′t(b)≥ exp(−π t/λ). �

Proof of Proposition 2.3. Notice that γ (s) = h−1(h(γ (0))+ s) for all s > T . By[Shapiro 1993, p. 162], for almost every ς ∈ ∂D the (possibly infinite) radial limith∗(ς)= limr→1 h(rς) exists and it is not a.e. constant on any subarc of ∂D.

If T >−∞, we know that the α-limit α(γ ) is a nonempty, compact, connectedsubset of ∂D. We want to show that it is a single point. If this were not thecase, α(γ ) would be a nontrivial subarc of ∂D. Hence, for each a ∈ α(γ ) apartfrom the endpoints, the radius from 0 to a would intersect γ infinitely often. Thusthere would exist sn ↘ T such that γ (sn) = rna. The existence of the radial limith∗(a)= limr→1 h(ra) implies

h∗(a)= limn

h(rna)= limn

h(γ (sn))= limn(h(z)+ sn)= h(z)+ T .


So h∗ would be a.e. constant on α(γ ), a contradiction. Therefore, α(γ ) must be asingle point; that is, the limit a := lims→T γ (s) ∈ ∂D exists.

Now we have to show that, given t > 0, a is not a fixed point. Notice thatlims→T + ϕt(γ (s)) = lims→T + h−1(h(z) + s + t) = h−1(h(z) + T + t) ∈ D. So,by Lindelof’s Theorem, limr→1 ϕt(ra) = h−1(h(z)+ T + t) ∈ D. In particular,limr→1 ϕt(ra) 6= a and a cannot be a fixed point. Clearly, h(z)+ T ∈ ∂�.

If T = −∞, the existence of the limit a := lims→−∞ γ (s) ∈ ∂D is guaranteedby [Shapiro 1993, p. 162] and we showed in Lemma 4.2 that a is a fixed point. �

Recall that the invariant set of � is

V (�)= int(⋂

t≥0

(�+ t)),

and that its connected components are denoted by V j (�), for j ∈ J . We nowsummarize the basic properties of the invariant set and its connected components.

Lemma 5.1. (1) If s ∈ R, then V (�)+ s = V (�).

(2) V j (�)+ s = V j (�) for all s ∈ R and all j ∈ J . In particular, each V j (�)

is a strip or a half-plane. Therefore, in the first case, there exist unique realnumbers a j and β j (�) such that

V j (�)={z ∈ C : a j −

12β j (�) < Im z < a j +

12β j (�)

},

and, in the second case, there is a real number a j such that

V j (�)= {z ∈ C : a j < Im z} or V j (�)= {z ∈ C : a j > Im z}.

Proof. (1) First suppose that s > 0. Then(⋂t≥0

(�+ t))

+ s =

⋂t≥0

(�+ t + s)=

⋂t≥s

(�+ t)=

⋂t≥0

(�+ t).

Therefore,

V (�)+ s = int((⋂

t≥0

(�+ t))

+ s)

= V (�).

Now, if s < 0, we have V (�)− s = V (�). So, V (�)+ s = V (�).

(2) By the first part of this lemma, we have V j (�) + s ⊆ V (�) + s = V (�).Moreover, V j (�)+ s is connected. So, there is k such that V j (�)+ s ⊆ Vk(�).Similarly, there is l such that Vk(�)− s ⊆ Vl(�). Therefore,

V j (�)+ s ⊆ Vk(�)⊆ Vl(�)+ s.

So, V j (�) = Vl(�) and V j (�)+ s = Vk(�). Why is k = j? Fix z ∈ V j (�) andconsider the curve τ ∈ [0, s] → σ(τ)= z + τ ∈ V (�). Clearly, σ([0, s]) is an arc


in V (�) (and so it is included in one connected component) that begins in V j (�)

and ends in Vk(�). Therefore V j (�)= Vk(�). �

Proof of Theorem 2.4. We notice that γz is defined for all t ∈ (−∞, 0] if and onlyif h(z)+ R ⊂�.

(1) Take z such that h(z) + R is in a half-plane V j (�) and consider the curves01(s)= h(z)− s and 02(s)= h(z)+ s for s > 0. We know, by the Denjoy–WolffTheorem, that lims→∞ h−1(02(s))= b. By Lemma 4.1, we have

lims→∞

h−1(01(s))= lims→∞

h−1(02(s))= b.

(2) Let z be a point of D such that 3 := h(z)+ R ⊂ �. Since, by assumption,lims→−∞ h−1(h(z) + s) = b, the union γz ∪ {b} is a Jordan curve contained inD ∪ {b} and satisfying ϕt(γz)= γz for all t ≥ 0. The curve γz divides the unit diskinto two connected components 21 and 22, where ∂21 = γz ∪ {b} and ∂22 =

∂D ∪ γz . Transferring this situation to � we have 3 = ∂h(21) ∩�. Therefore,3⊂ ∂(h(21)+t)∩� for all t ≥ 0. That is, γz ⊂ ∂ϕt(21) for all t ≥ 0. On the otherhand, the restriction of ϕt to 21 (with the obvious extension to the point b) is ahomeomorphism. So, ϕt(21) is a Jordan domain whose boundary contains the setγz . Therefore, ϕt(21)=21. Again, going over to �, we have h(21)+ t = h(21)

for all positive t , and a posteriori, for all t ∈ R. That is, h(21) is a strip or ahalf-plane and it is contained in V .

Suppose that h(21) is a strip bounded by two lines. One of them is 3. Let ϒdenote the other line of the boundary of h(21). There exist β > 0 and a ∈ Ri suchthat the map

σ(w)=1 − ew/βe−a/β

1 + ew/βe−a/β

is a bijection from h(21) onto the unit disc. Notice that σ has a continuous exten-sion σ to h(21) (with image D\ {−1, 1}). By the Caratheodory Theorem (see, forexample, [Milnor 1999, Theorem 17.16]) and since21 is a Jordan domain, we haveσ ◦h|21. can be extended to a homeomorphism from 21 onto D. Given w ∈ϒ andtaking a sequence (wn) in h(21) that converges to w, it is not difficult to prove that(h−1(wn)) converges to b. So, σ ◦ h|21(h

−1(wn)) converges to σ ◦ h|21(b) ∈ ∂D.On the other hand, σ ◦ h|21(h

−1(wn)) = (σ (wn)) converges to σ (w). That is,σ is constant on ϒ . This is a contradiction, since w 7→ ew/β is not constant onany horizontal line. Therefore, h(21) is a half-plane contained in V (�) and weconclude that there is a half-plane V j (�) such that h(21) ⊂ V j (�). In particular,h(z) ∈ h(21)⊂ V j (�). �

Proof of Theorem 2.5. (1) When we have a strip V j (�) and two half-lines inV j (�), say h(z1)+(−∞, 0] and h(z2)+(−∞, 0], we can apply Lemma 4.1 (join-ing the points h(z1) and h(z2) by a segment) to get lims→−∞ h−1(h(z1)+ s) =


lims→−∞ h−1(h(z2) + s). So, the fixed point ξ(V j (�)) associated to V j (�) iswell-defined.

We now check that ξ(V j (�)) is repulsive. We have to verify that

1< ϕ′

t(ξ(V j (�))) <+∞.

Take zn = h−1(a j i − n). Then

ρD(zn, ϕt(zn))= ρ�(a j i − n, a j i − n + t)≤ 2∫

[a j i−n,a j i−n+t]

|dw|

δ�(w)≤

4tβ j (�)

.

Therefore, there is a constant σ < 1 such that the pseudo-hyperbolic distance be-tween zn and ϕt(zn) is uniformly bounded by σ . That is,∣∣∣∣ zn −ϕt(zn)

1 − znϕt(zn)

∣∣∣∣≤ σ for all n.

Moreover, for n large enough we have |zn| ≥ σ . From this, and using [Shapiro1993, Exercise 1, p. 73], we conclude that

1 − |ϕt(zn)|

1 − |zn|≤

1 + σ

1 − σ.

Hence,

lim infz→ξ(V j (�))

1 − |ϕt(z)|1 − |z|

≤1 + σ

1 − σ.

The Julia–Caratheodory Theorem implies that ϕ′t(ξ(V j (�)))≤ (1 +σ)/(1 −σ) <

+∞. To prove that 1 < ϕ′t(ξ(V j (�))), we will show that ξ(V j (�)) is not the

Denjoy–Wolff point of the semigroup (ϕt). If this were the case, we would haveα(h−1(0))= b, where 0 = {a j i}+R. By Theorem 2.4, 0 would lie in the closureof one of the half-planes of V (�): a contradiction because 0 is in one of the stripsof V (�).

(2) We begin by showing that the map ξ that sends a strip to its associated fixedpoint is injective. Consider two components V j (�) and Vl(�) such that ξ(V j (�))=

ξ(Vl(�)) = ξ and assume that V j (�) 6= Vl(�). Take 0 j = {a j i} + R and 0l =

{al i}+R. Set γ j = h−1(0 j ) and γl = h−1(0l). The curves γ j and γl are disjoint inD and connect ξ to the Denjoy–Wolff point b; thus they bound a simply connectedregion 2 ⊂ D such that ∂2 = γ j ∪ γl ∪ {b, ξ}. Fix t > 0. We have ϕt(γ j ) = γ j

and ϕt(γl) = γl . Going over to � we have 0 j ∪ 0l ⊂ ∂(h(2)+ t) ∩�; that is,γ j ∪γl ⊂ ∂ϕt(2). On the other hand, the restriction of ϕt to2 (with the obvious ex-tensions to the points b and ξ ) is a homeomorphism. So, ϕt(2) is a Jordan domainwhose boundary contains the Jordan curve γ j ∪γl ∪{b, ξ}. Therefore, ϕt(2)=2.In particular, h(2) lies in V (�) and is included in a connected component of V ,which is a contradiction since h(2)∩V j (�) 6=∅ and h(2)∩Vl(�) 6=∅. Therefore,the map ξ is injective.


Now, we will see that the map ξ is onto. Let ξ ∈ ∂D be a fixed point of ϕt with1<ϕ′

t(ξ)<∞. Take ε as the positive constant associated to the fixed point ξ givenin Lemma 4.4.

First, we are going to show that there are a strip V j (�) and 0< r0 < 1 such that⋃r≥r0

B(h(rξ), ε)⊂ V j (�).

Let 3= h([0, 1)ξ) and 3n =3+n. The curve 3 connects h(0) to ∞. Moreover,there are two constants M1 and M2 such that M1 ≤ Im h(rξ) ≤ M2 for all r andlimr→1− Re h(rξ) = −∞ (otherwise, by Lemma 4.1, ξ would be the Denjoy–Wolff point). So, for each n, there is ηn = h(rnξ)+ n ∈3n such that Re ηn = −1.Let (ηnk ) be a subsequence of (ηn) converging to a number η, with Re η = −1,M1 ≤ Im η ≤ M2 and |ηnk − η|< ε/2 for all k. The choice of ε implies that

δ�(h(rξ)+ n)≥ δ�(h(rξ))≥ ε

for all r . In particular,

B(η, ε/2)⊆ B(ηnk , ε)= B(h(rnkξ), ε)+ nk ⊂�+ nk

for all k. Therefore, B(η, ε/2) ⊂ V (�). Let V j (�) be the connected componentof V (�) containing B(η, ε/2). Notice that ηnk ∈ B(η, ε/2)⊂ V j (�). Suppose thatthere is no r0 such that ⋃

r≥r0

B(h(rξ), ε)⊂ V j (�).

Then, for infinitely many k, we can find zk with inf{|zk −w| : w ∈ 3nk } < ε/2,Re zk<−1 and zk ∈∂V j (�). Takewk =h(skξ)+nk ∈3nk such that |wk−zk |<ε/2.We have

B(zk, ε/2)⊆ B(wk, ε)= B(h(skξ), ε)+ nk ⊂�+ nk

for all k. Now choose mk such that zk = zk + mk satisfies −2 ≤ Re(zk)≤ −1 and,passing to a subsequence (still written the same), we consider that (zk) convergesto a point ζ . Of course, we can also suppose that |zk − ζ |< ε/4 for all k, so

B(ζ, ε/4)⊆ B(zk, ε/2)⊂ B(h(skξ), ε)+ nk + mk ⊂�+ nk + mk

for all k. Therefore, B(ζ, ε/4)⊂ V (�). Let Vl(�) be the connected component ofV (�) containing B(ζ, ε/4). Then zk ∈ Vl(�), which contradicts the fact that zk isin ∂V j (�). That is, ⋃

r≥r0

B(h(rξ), ε)⊂ V j (�)

for r0 sufficiently close to 1.Next, we show that V j (�) is a strip. Suppose V j (�) is a half-plane, say V j (�)=

{z ∈ C : a j < Im z}, and take 01(s) = (a j + 1)i − s and 02(s) = (a j + 1)i + s for


s> 0. Then, by Lemma 4.1, lims→∞ h−1(01(s))= lims→∞ h−1(02(s))= b, whichcontradicts the inequality 1<ϕ′

t(ξ) (recall that b is the Denjoy–Wolff point of ϕt ).Now we prove that ξ(V j (�))= ξ . Take 01(s)= a j i − s for s > 0 and 02(r)=

h(rξ) for r < 1. We know that

lims→∞

h−1(01(s))= ξ(V j (�)) and limr→1

h−1(02(r))= ξ.

We check that the limits coincide. If there is a sequence (wn) in 01[0,∞)∩02[0, 1)such that wn → ∞, clearly lims→∞ h−1(01(s))= limr→1 h−1(02(r)). Otherwise,using Lemma 4.1 again and bearing in mind that 01 and 02(r) belong to V j (�)

for r large enough, we deduce that lims→∞ h−1(01(s))= limr→1 h−1(02(r)). �

The following lemma is well-known.

Lemma 5.2. Given ξ ∈ ∂D and a sequence (zn) in D that converges to ξ , thefollowing assertions are equivalent.

(1) There exists α = limn Arg(1 − ξ zn).

(2) There exists m = limn1−ξ zn|1−ξ zn|

.

(3) There exists µ= limnIm ξ zn

1−Re ξ zn.

Moreover, if these assertions are satisfied, eiα= m and µ= −tanα.

Lemma 5.3. Suppose given ξ ∈ ∂D, a repulsive boundary fixed point of thesemigroup of analytic functions (ϕt) with associated strip V j (�), and z ∈ D withh(z) ∈ V j (�). Then Slope−(γz) is a single point of

(−π2 ,

π2

).

Proof. Given s > t > 0, by the Invariant Schwarz–Pick Lemma [Shapiro 1993,p. 60] applied to the function ϕs−t , we have∣∣∣∣ ϕs+1(w)−ϕs(w)

1 −ϕs+1(w)ϕs(w)

∣∣∣∣≤ ∣∣∣∣ ϕt+1(w)−ϕt(w)

1 −ϕt+1(w)ϕt(w)

∣∣∣∣for all w ∈ D. In particular, taking the point w = h−1(h(z) − s − t) (which iswell-defined because h(z) ∈ V j (�)) we obtain∣∣∣∣ h−1(h(z)− t + 1)− h−1(h(z)− t)

1 − h−1(h(z)− t + 1)h−1(h(z)− t)

∣∣∣∣≤ ∣∣∣∣ h−1(h(z)− s + 1)− h−1(h(z)− s)

1 − h−1(h(z)− s + 1)h−1(h(z)− s)

∣∣∣∣.That is, the function f defined by

f (t)=∣∣∣∣ h−1(h(z)−t+1)−h−1(h(z)−t)

1−h−1(h(z)−t+1)h−1(h(z)−t)

∣∣∣∣= ∣∣∣∣ ξh−1(h(z)−t+1)−ξh−1(h(z)−t)

1−ξh−1(h(z)−t+1)ξh−1(h(z)−t)

∣∣∣∣


is increasing. Let l = limt→∞ f (t). Since

h−1(h(z)− t + 1)− h−1(h(z)− t)

1 − h−1(h(z)− t + 1)h−1(h(z)− t)∈ D,

for all t , we have l ≤ 1.Consider the curve

γ (t)=Im ξh−1(h(z)− t)

1 − Re ξh−1(h(z)− t),

with t > 0, and take a sequence (tn) increasing to infinity such that γ (tn) goes toµ. By Proposition 4.5, the sequence (h−1(h(z)− tn)) converges nontangentially tothe point ξ . So, by Lemma 5.2, there exists α = limn Arg(1 − ξh−1(h(z)− tn)) ∈(−π2 ,

π2

)andµ=− tanα∈R. Set un =Re ξh−1(h(z)−tn), vn = Im ξh−1(h(z)−tn),

an = Re ξh−1(h(z) − tn + 1), and bn = Im ξh−1(h(z) − tn + 1). We know thatvn/(1 − un) converges to µ. Notice that un → 1 and vn → 0. Moreover,

limt→∞

1 − ξϕ1(h−1(h(z)− tn))1 − ξh−1(h(z)− tn)

= ϕ′

1(ξ).

In particular,

limn

1 − (an + ibn)

1 − (un + ivn)= ϕ′

1(ξ) > 1.

A brief calculation shows that limn(1−an)/(1−un)=ϕ′

1(ξ) and limn bn/(1−un)=

µϕ′

1(ξ). On the other hand, using the definition of f (t), we have∣∣1 − (an − ibn)(un + ivn)∣∣ f (tn)=

∣∣(an + ibn)− (un + ivn)∣∣.

Dividing by 1 − un , we get∣∣∣∣1+un1−an

1−un−

bn

1−unvn−

vn

1−unani +

bn

1−ununi∣∣∣∣ f (tn)

=

∣∣∣∣1−1−an

1−un+

bn

1−uni −

vn

1−uni∣∣∣∣.

Taking limits, we have∣∣1+ϕ′

1(ξ)−µi +µϕ′

1(ξ)i∣∣ l =

∣∣1−ϕ′

1(ξ)+µϕ′

1(ξ)i −µi∣∣,

that is, ∣∣∣∣1 +ϕ′

1(ξ)

1 −ϕ′

1(ξ)−µi

∣∣∣∣ l = |1 −µi |.

Therefore,

µ2(1 − l2)=

(1 +ϕ′

1(ξ)

1 −ϕ′

1(ξ)

)2

l2− 1.

Since the fraction in parentheses lies in (−∞,−1), we see that l 6= 1 and µ hasat most two values. Therefore, by Lemma 5.2, the set Slope−(γz) has at most two


points, and being an interval, it must be a single point, which must be the numberα found above. So, Slope−(γz) is a single point of

(−π2 ,

π2

). �

Proof of Theorem 2.6. Fix t > 0. By Proposition 3.3, we have

logϕ′

t(ξ)= limr→1

ρD(rξ, ϕt(rξ))

= minθ∈(− π

2 ,π2 )

{lim

nρD(zn, ϕt(zn)) : zn → ξ, lim

nArg(1 − ξ zn)= θ

}.

By [Poggi-Corradini 2000, Corollary 1.5], given θ ∈(−π2 ,

π2

), there is a sequence

(zn) in D such that zn =ϕt(zn+1), zn → ξ and Arg(1− ξ zn)→ θ . By Theorem 2.5,given such a sequence, there is a point c ∈ V j (�) such that zn = h−1(c − nt). Wecheck that c ∈ V j (�). If this were not so, clearly lims→+∞ δ�(c − s)= 0, leadingto a contradiction, since on the one hand

ρD(zn, ϕt(zn))= ρ�(c − nt, c − nt + t)

≥12 log

(1 +

tmin{δ�(c − nt), δ�(c − nt + t)}

),

which tends to ∞, while on the other, by Proposition 3.3 and since θ ∈(−π2 ,

π2

),

we have

limn→∞

ρD(zn, ϕt(zn))= log|ϕ′

t(ξ)+ e−2iθ| + |ϕ′

t(ξ)− 1|

|ϕ′t(ξ)+ e−2iθ | − |ϕ′

t(ξ)− 1|<∞.

Conversely, given c ∈ V j (�) and taking zn = h−1(c − nt) for all n, by Lemma5.3, there is θ ∈

(−π2 ,

π2

)such that limn Arg(1 − ξ znk )= θ . Therefore,

log(ϕ′

t(ξ))= minθ∈(− π

2 ,π2 )

{lim

nρD(zn, ϕt(zn)) : zn → ξ, lim

nArg(1 − ξ zn)= θ

}= min

c∈V j (�)

{lim

nρD(zn, ϕt(zn)) : zn = h−1(c − nt)

}.

Thus, to calculate logϕ′t(ξ), we will take c ∈ V j (�), evaluate limn ρD(zn, ϕt(zn)),

where zn = h−1(c − nt), and take the minimum of that limit. So fix an arbitrary cin V j (�). For all n ∈ N, set pn = h−1(c − nt) and let αpn be an automorphism ofthe unit disc such that αpn (0)= pn . Consider the function fn(z)= h ◦αpn (z)+nt .Then fn(0)= c and fn(D)= h(D)+nt =�+nt ⊆�, for all n. Thus { fn : n ∈ N} isa normal family, and so has a subsequence fnk converging uniformly on compactsubsets of D to an analytic function f on D. Also, fn+1(D) ⊂ fn(D) for all n, sof (D)⊂ ∩n fn(D)= ∩n(�+nt)⊂ V (�). Since f (0)= c, we have f (D)⊆ V j (�).Moreover, f −1

nk: V j (�) → D is also a normal family. So, it has a subsequence

(still written the same) that converges uniformly on compact subsets of V j (�) toan analytic function g : V j (�) → D. Since g(c) = 0, we have g(V j (�)) ⊆ D. If


z ∈ D and k0 is large enough, then

{ fnk (z) : k ≥ k0} ∪ { f (z)}

is a compact subset of V j . Thus z = f −1nk( fnk (z))→ g( f (z)); that is, z = g( f (z)).

Similarly, w= f (g(w)) for all w ∈ V j (�). Thus, f is one-to-one, f (D)= V j (�),f −1

= g, and f −1nk

converges uniformly on compact subsets of V j (�) to f −1. Asimple calculation shows that

f −1n (w)= α−1

pn(h−1(w− nt))

for all w. So, given z ∈ D,

f −1n ( fn(z)+ t)= α−1

pn(h−1( fn(z)+ t − nt))

= α−1pn(h−1(h ◦αpn (z)+ t))= α−1

pn◦ϕt ◦αpn (z).

In particular, taking z = 0, we have αpn ( f −1n (c + t))= ϕt(pn).

Recall that

9(z)=β j

πLog

1 − z1 + z

+ (Re c + a j i)

is another Riemann map of V j (�). Thus, there is θ ∈ [0, 2π ] and d ∈ D such thatf −1

◦9(z)= eiθ (z − d)/(1 − dz) for all z ∈ D. Since 9(d)= f (0)= c, we get

d =1 − e(π/β j (�)) Im ci e−(π/β j (�)) a j i

1 + e(π/β j (�)) Im ci e−(π/β j (�)) a j i.

Now set u = e(π/β j (�)) Im ci e−(π/β j (�)) a j i and v = e(π/β j (�)) t . We obtain

limn→∞

ρD(pnk , ϕt(pnk ))

= limn→∞

ρD(αpnk(0), αpnk

( f −1nk(c + t)))

= limn→∞

ρD(0, f −1nk(c + t))= ρD(0, f −1(c + t))

= ρD

(0, eiθ 9

−1(c + t)− d1 − d9−1(c + t)

)= ρD

(0,9−1(c + t)− d1 − d9−1(c + t)

)= ρD

(0,

1−uv1+uv −

1−u1+u

1 −u−1u+1

1−uv1+uv

)= ρD

(0, u

1 − v

1 + u2v

)= ρD

(0,

1 − v

1 + u2v

).

Bearing in mind that

d(

0,1−v

1+u2v

)=

∣∣∣∣ 1−v

1+u2v

∣∣∣∣= v−1|1+u2v|

=v−1√

1+v2+2v cos(2π(Im c−a j )/β j (�))


and that v does not depend on c, we have

logϕ′

t(ξ)= minc∈V j (�)

{lim

nρD(zn, ϕt(zn)) : zn = h−1(c − nt)

}= min

c∈V j (�)ρD(0,

1 − v

1 + u2v)= log

1 + minc∈V j (�) d(

0, 1−v

1+u2v

)1 − minc∈V j (�) d

(0, 1−v

1+u2v

)

= log

1 + minc∈V j (�)

v− 1√1 + v2 + 2v cos(2π(Im c − a j )/β j (�))

1 − minc∈V j (�)

v− 1√1 + v2 + 2v cos(2π(Im c − a j )/β j (�))

= log1 +

v−1v+1

1 −v−1v+1

=π

β j (�)t.

That is, ϕ′t(ξ(V j (�)))= exp(π t/β j (�)). �

Proof of Theorem 2.7. Take z such that

h(z) ∈(⋂

t≥0(�+ t))\(⋃

j∈J V j (�)).

Then, by Proposition 2.3, α(γz) is a single point ξ which is a fixed point. ByTheorems 2.4 and 2.5, ξ is neither attractive nor repulsive. So, it must be a super-repulsive boundary fixed point of 8.

Suppose that there is another point z such that γz 6= γz and α(γz) = {ξ}. Thenγz and γz are disjoint curves in D connecting ξ to b. So, they bound a simplyconnected region 2 ⊂ D such that ∂2 = γz ∪ γz ∪ {ξ, b}. Let us fix t > 0. Wehave that ϕt(γz) = γz and ϕt(γz) = γz . Passing to �, we have h(γz) ∪ h(γz) ⊂

∂(h(2)+ t)∩�; that is, γz ∪ γz ⊂ ∂ϕt(2). On the other hand, the restriction ofϕt to 2 (with the obvious extensions to the points ξ and b) is a homeomorphism.Thus ϕt(2) is a Jordan domain whose boundary contains the Jordan curve γz ∪γz ∪

{ξ, b}. Therefore, ϕt(2)=2. In particular, h(2) lies in V (�) and is included in aconnected component of V (�), which is a contradiction since h(z) /∈

⋃j∈J V j (�).

�

Lemma 5.4. Suppose (zn) and (wn) are two sequences in the unit disk that con-verge to ξ ∈ ∂D and such that

(1) Arg(1 − ξ zn)→ α and Arg(1 − ξwn)→ β with α, β ∈[−π2 ,

π2

]and

(2) ρD(zn, wn)→ 0.

Then α = β.


Proof. We have zn = (1 − rneiθn )ξ and wn = (1 − sneiξn )ξ with θn, ξn ∈(−π2 ,

π2

).

Notice that {n ∈ N : rn ≥ sn} or {n ∈ N : sn ≥ rn} is infinite. Without loss ofgenerality, we assume that rn ≥ sn for all n. Again, taking a subsequence (stillwritten the same), we suppose that sn/rn → λ ∈ [0, 1]. By Proposition 3.2, wehave ∣∣∣∣ zn −wn

1 − znwn

∣∣∣∣→ 0.

Moreover,∣∣∣∣ zn −wn

1 − znwn

∣∣∣∣= ∣∣∣∣ −rneiθn + sneiξn

rne−iθn + sneiξn − rnsnei(ξn−θn)

∣∣∣∣= ∣∣∣∣ (sn/rn) ei(ξn−θn) − 11 + (sn/rn) ei(ξn+θn) − sneiξn

∣∣∣∣.If α+β=±π , then α=β. Otherwise, 1+(sn/rn) ei(ξn+θn)−sneiξn → 1+λei(β+α),which is nonzero. So,∣∣∣∣ (sn/rn) ei(ξn−θn) − 1

1 + (sn/rn) ei(ξn+θn) − sneiξn

∣∣∣∣→ ∣∣∣∣λei(β−α)− 1

1 + λei(β+α)

∣∣∣∣= 0.

Therefore, we obtain that 1 = λei(β−α) and we conclude that α = β. �

Proof of Theorem 2.8. The proof that given z ∈ D, the set ω(Arg(1 − bϕt(z))) isa single point is similar to that of Lemma 5.3. So, Slope+(γz) is a single point.Once we have obtained this, by Lemma 4.3, we see that Slope+(γz) ∈

(−π2 ,

π2

).

Now consider the map z 7→ k(z) = limt→∞ Arg(1 − bϕt(z)). Assume for themoment the following two claims (which we will prove below).

Claim 1. The function k is continuous. Therefore, k(D) is an interval.

Claim 2. If k(z1)= k(z2), then Im h(z1)= Im h(z2).

By Lemma 4.4, there is ε > 0 such that δ�(h(rξ)) ≥ ε for all r ∈ [0, 1). Takethe real number a such that

�⊆2 :={z ∈ C : a −

12ν(�) < Im z < a +

12ν(�)

}.

Fix M > 0. Recall that the set {z ∈ D : ρD(z, (−1, 1))≤ M} is a lens associated toa certain angle 0< α < π

2 .We show that there are points z such that the corresponding trajectory γz satisfy

that limt→+∞ ρD(γz(t), (−1, 1)) > M . Choose ε > δ > 0 with log(1+(ε−δ)/δ) >

4M , take z1, z2 ∈ D with Im h(z1) > a +12ν(�)−δ and Im h(z2) < a −

12ν(�)+δ,

and take t > 0. Then δ�(h(zi ) + t) ≤ δ for i = 1, 2. By the Distance Lemma(Proposition 3.5), we have


ρD(ϕt(zi ), (−1, 1))≥ inf0<r<1

ρ�(h(zi )+ t, h(r))

≥ infs>0

12 log

(1 +

|h(zi )+ t − h(r)|min{δ�(h(zi )+ t), δ�(r)}

)≥

12 log

(1 +

ε− δ

δ

)≥ 2M.

Therefore, |Slope+(γzi )|>α for i = 1, 2. Moreover, Slope+(γz1) and Slope+(γz2)

have different signs, for the following reason: let c be a point of � such thatthere is a sequence rn ↗ 1 with Im h(rnξ)→ Im c. Then, by Lemma 5.4, k(c) =

Slope+(γc) = 0. Of course, we may assume that Re c = Re h(z1) = Re h(z2)

and that [h(z1), h(z2)] ⊂ �. By Claim 2, the map s ∈ [Im h(z2), Im h(z1)] 7→

k(h−1(Re c + is)) is monotone. Therefore, k(z1) ≥ k(0) ≥ k(z2) (or vice versa).This and Claim 1 imply that (−α, α)⊂ k(D) for all α. Thus k(D)=

(−π2 ,

π2

).

Proof of Claim 1. Take zn → z. We have to check that k(zn) → k(z). Since k isconstant on each trajectory, we may assume that, for n large enough, Re h(zn) =

Re h(z) and the segment [h(zn), h(z)] is in �. We have

ρD(ϕt(zn), ϕt(z))

= ρ�(h(zn)+ t, h(z)+ t)≤ 2∫

[h(zn)+t,h(z)+t]

ds(x)δ�(x)

≤ 2|h(zn)+ t − h(z)− t |

min{δ�(h(zn)+ t), δ�(h(z)+ t)}≤ 2

|h(zn)− h(z)|min{δ�(h(zn)), δ�(h(z))}

.

Since h(zn) converges to h(z) and δ�(h(zn)) converges to δ�(h(z)), which isnonzero, we obtain that ρD(ϕt(zn), ϕt(z)) → 0. Now, take tn → +∞ such thatϕtn (zn) = (1 − rneiθn )b with rn ≤

1n , and |θn − k(zn)| ≤

1n for all n. Without

loss of generality, we may assume that k(zn) → β ∈[−π2 ,

π2

]. Now consider

ϕtn (z) = (1 − sneiξn )b. It is clear that sn → 0 and ξn → k(z). By Lemma 5.4, wehave β = k(z). That is, k(zn)→ k(z). �

Proof of Claim 2. Take two orbits {ϕt(z1) : t ≥ 0} and {ϕt(z2) : t ≥ 0} and pointsan = (1 − rneiθn )b on the orbit {ϕt(z1) : t ≥ 0} and bn = (1 − rneiξn )b on the orbit{ϕt(z2) : t ≥ 0}, such that θn → α, ξn → α, α ∈

(−π2 ,

π2

), and rn → 0. Then∣∣∣∣ an − bn

1 − anbn

∣∣∣∣= ∣∣∣∣ −eiθn + eiξn

e−iθn + eiξn − rnei(ξn−θn)

∣∣∣∣ →n→∞

0.

That is, ρD(an, bn) tends to 0. Setting h(an)= h(z1)+ tn and h(bn)= h(z2)+ sn ,we have

ρD(an, bn)= ρ�(h(an), h(bn))= ρ�(h(z1)+ tn, h(z2)+ sn)

≥12 log

(1 +

|h(z1)+ tn − h(z2)− sn|

min{δ�(h(z1)+ tn), δ�(h(z2)+ sn)

}).


Moreover, min{δ�(h(z1)+ tn), δ�(h(z2)+ sn)} is bounded below, so the sequence(|h(z1)+ tn − h(z2)− sn|

)must go to zero. But∣∣Im(h(z1)− h(z2))

∣∣≤ ∣∣h(z1)+ tn − h(z2)− sn∣∣.

Therefore, Im h(z1)= Im h(z2), proving the claim and Theorem 2.8. �

Proof of Theorem 2.9. (1) Take z1, z2 ∈ D and θ ∈ Slope+(γz1). Then there is(tn)↗ ∞ such that Arg

(1− bh−1(h(z1)+ tn)

)→ θ . Bearing in mind Lemma 5.4,

to obtain that θ ∈ Slope+(γz2), it is enough to show that

limn→∞

ρ�(h(z1)+ tn, h(z2)+ Re(h(z1)− h(z2))+ tn

)= 0.

First notice we can assume that Re h(z1) = Rew. We estimate the hyperbolicdistance by integrating along the segment

S =[h(z1)+ tn, h(z2)+ Re(h(z1)− h(z2))+ tn

],

which is contained in � when n is large enough:


)≤ 2

∫S

ds(z)δ�(z)

≤ 2 |h(z1)− h(z2)| maxz∈S

1δ�(z)

.

If we take n large enough, we have

maxz∈S

1δ�(z)

=1

min{δ�(h(z1)+ tn), δ�(h(z2)+ Re(h(z1)− h(z2))+ tn)

}≤

1min

{δ�(w+tn)− |h(z1)−w|, δ�(w+tn)− |h(z2)+ Re(h(z1)−h(z2))−w|

}=

1δ�(w+ tn)− max

{|h(z1)−w|, |h(z2)+ Re(h(z1)− h(z2))−w|

} ,which tends to 0 as n goes to ∞. That is,

limn→∞


)= 0.

(2) Since Slope+(γz) is an interval, it is enough to apply Lemma 4.6. �

Proof of Theorem 2.10. (1) Recall that Slope−(γz) is a nonempty, compact andconnected subset of

[−π2 ,

π2

]. Moreover, it is clear that lims→−∞ δ�(h(z)−s)<∞.

So, by Lemma 4.6, Slope−(γz) is equal to −π2 or π

2 .


(2) Arguing as in the proof of Theorem 2.8, the function

w 7→ k(w)= lims→∞

Arg(1 − bh−1(w− s))

is continuous as w runs over one of the half-planes of the invariant set. Since k(w)is either −

π2 or π

2 for all w, it must be constant. �

Proof of Theorem 2.11. (1) It is enough to apply Lemma 5.3 and [Poggi-Corradini2000, Corollary 1.5].

(2) Recall that Slope−(γz) is a nonempty, compact, connected subset of[−π2 ,

π2

].

Moreover, it is clear that lims→+∞ δ�(h(z)− s)= 0. On the one hand,

ρD

(h−1(h(z)− s), ϕt(h−1(h(z)− s))

)= ρ�

(h(z)− s, h(z)− s + t

)≥

12 log

(1 +

tmin{δ�(h(z)− s), δ�(h(z)− s + t)}

),

which tends to ∞ with s. On the other hand, if there is (sn)↗ ∞ such that

limn→∞

Arg(1 − ξ(V j (�))h−1(h(z)− sn)

)= θ ∈

(−π2 ,

π2

),

by Proposition 3.3, we have

limn→∞

ρD(h−1(h(z)− sn), ϕt(h−1(h(z)− sn)))

= log|ϕ′

t(ξ(V j (�)))+ e−2iθ| + |ϕ′

t(ξ(V j (�)))− 1|

|ϕ′t(ξ(V j (�)))+ e−2iθ | − |ϕ′

t(ξ(V j (�)))− 1|<∞,

yielding a contradiction. That is, Slope−(γz) is either π2 or π

2 . �

Acknowledgement

The works of Poggi-Corradini [1998; 2000] on discrete iteration of analytic self-maps of the unit disk have been an important source of inspiration for the mathe-matical techniques developed in this paper.

References

[Berkson and Porta 1978] E. Berkson and H. Porta, “Semigroups of analytic functions and compo-sition operators”, Michigan Math. J. 25:1 (1978), 101–115. MR 58 #1112 Zbl 0382.47017

[Cowen 1981] C. C. Cowen, “Iteration and the solution of functional equations for functions analyticin the unit disk”, Trans. Amer. Math. Soc. 265:1 (1981), 69–95. MR 82i:30036 Zbl 0476.30017

[Cowen and MacCluer 1995] C. C. Cowen and B. D. MacCluer, Composition operators on spacesof analytic functions, CRC Press, Boca Raton, FL, 1995. MR 97i:47056 Zbl 0873.47017


[Cowen and Pommerenke 1982] C. C. Cowen and C. Pommerenke, “Inequalities for the angularderivative of an analytic function in the unit disk”, J. London Math. Soc. (2) 26:2 (1982), 271–289.MR 84a:30006 Zbl 0476.30001

[Heins 1981] M. Heins, “Semigroups of holomorphic maps of a Riemann surface into itself whichare homomorphs of the set of positive reals considered additively”, pp. 314–331 in E. B. Christoffel:The influence of his works on mathematics and the physical sciences (Aachen/Monschau, 1979),edited by P. L. Butzer and F. Feher, Birkhäuser, Basel, 1981. MR 84f:30012 Zbl 0474.30022

[Milnor 1999] J. Milnor, Dynamics in one complex variable: introductory lectures, Vieweg, Braun-schweig, 1999. MR 2002i:37057 Zbl 0946.30013

[Needham and King 1994] D. J. Needham and A. C. King, “On meromorphic complex differentialequations”, Dynam. Stability Systems 9:2 (1994), 99–122. MR 95m:34010 Zbl 0813.34005

[Poggi-Corradini 1998] P. Poggi-Corradini, “Angular derivatives at boundary fixed points for self-maps of the disk”, Proc. Amer. Math. Soc. 126:6 (1998), 1697–1708. MR 98g:30049 Zbl 0891.30014

[Poggi-Corradini 2000] P. Poggi-Corradini, “Canonical conjugations at fixed points other than theDenjoy–Wolff point”, Ann. Acad. Sci. Fennicæ Math. 25:2 (2000), 487–499. MR 2001f:30033Zbl 0958.30012

[Pommerenke 1975] C. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen,1975. MR 58 #22526 Zbl 0298.30014

[Pommerenke 1992] C. Pommerenke, Boundary behaviour of conformal maps, Grundlehren derMathematischen Wissenschaften 299, Springer, Berlin, 1992. MR 95b:30008 Zbl 0762.30001

[Shapiro 1993] J. H. Shapiro, Composition operators and classical function theory, Springer, NewYork, 1993. MR 94k:47049 Zbl 0791.30033

[Shoikhet 2001] D. Shoikhet, Semigroups in geometrical function theory, Kluwer Academic Pub-lishers, Dordrecht, 2001. MR 2002g:30012 Zbl 0980.30001

[Siskakis 1985] A. G. Siskakis, Semigroups of composition operators and the Cesàro operator onH p(D), Ph.D. thesis, University of Illinois, 1985.

Received March 31, 2004. Revised November 4, 2005.

MANUEL D. CONTRERAS

CAMINO DE LOS DESCUBRIMIENTOS, S/N

DEPARTAMENTO DE MATEMATICA APLICADA IIESCUELA SUPERIOR DE INGENIEROS

UNIVERSIDAD DE SEVILLA

41092, SEVILLA

SPAIN

[email protected]

SANTIAGO D IAZ-MADRIGAL

CAMINO DE LOS DESCUBRIMIENTOS, S/N

DEPARTAMENTO DE MATEMATICA APLICADA IIESCUELA SUPERIOR DE INGENIEROS

UNIVERSIDAD DE SEVILLA

41092, SEVILLA

SPAIN

[email protected]


UNCOUNTABLY MANY INEQUIVALENT LIPSCHITZHOMOGENEOUS CANTOR SETS IN R3

DENNIS GARITY, DUŠAN REPOVŠ AND MATJAŽ ŽELJKO

General techniques are developed for constructing Lipschitz homogeneouswild Cantor sets in R3. These techniques, along with Kauffman’s versionof the Jones polynomial and previous results on Antoine Cantor sets, areused to construct uncountably many topologically inequivalent such wildCantor sets in R3. This use of three-dimensional finite link invariants todetect distinctness among wild Cantor sets is unexpected. These Cantorsets have the same Antoine graphs and are Lipschitz homogeneous. As acorollary, there are uncountably many topologically inequivalent Cantorsets with the same Antoine graph.

1. Introduction

Malesic and Repovs [1999] have constructed a specific example of a wild Cantorset in R3 that is Lipschitz homogeneously embedded. This answered negatively aquestion in [Repovs et al. 1996] as to whether Lipschitz homogeneity of a Cantorset implied tameness. In this paper, we introduce more general techniques fordetecting the Lipschitz homogeneity of Cantor sets in Rn . These techniques allowus to construct uncountably many topologically distinct Lipschitz homogeneouswild Cantor sets in R3. These Cantor sets are all simple Antoine Cantor sets withthe same Antoine graph as defined in [Wright 1986]. The fact that the constructedCantor sets are all topologically distinct is a consequence of a result of Sher [1968]and a computation of Kauffman’s version [1988] of the Jones polynomial for thecenter lines of certain tori used in the construction. It is hoped that the techniquesin this paper may also prove to be applicable to showing that certain Blankinshiptype Cantor sets [Blankinship 1951; Eaton 1973] in Rn for n ≥4 can be constructedso as to be Lipschitz homogeneously embedded.

MSC2000: primary 54E45, 54F65; secondary 57M30, 57N10.Keywords: wild Cantor set, Lipschitz homogeneity, similitude, coefficient of similarity, defining

sequence, link invariant.Garity was supported in part by NSF grants DMS 0139678 and DMS 0104325. Repovš and Matjažwere supported in part by MESS grant 0101-509. All authors were supported in part by MESS grantSLO-US 2002/01.

287

288 DENNIS GARITY, DUŠAN REPOVŠ AND MATJAŽ ŽELJKO

2. Notation and Background

Lipschitz maps and similitudes. A map S : Rn→ Rn is said to be a Lipschitz map

if there exists a constant λ such that

|S(x)− S(y)| ≤ λ|x − y| for every x, y ∈ Rn

and the smallest such λ is called the Lipschitz constant of S. In the special casewhen

|S(x)− S(y)| = λ|x − y| for every x, y ∈ Rn

the map S is called a similarity and the number λ is called the coefficient of simil-itude. Finally, when λ= 1 the map S is called an isometry.

A Cantor set C in R3 is Lipschitz homogeneously embedded if for each pair ofpoints x and y in C there is a Lipschitz homeomorphism h : Rn

→ Rn with h−1

also Lipschitz such that h(C)= C and h(x)= y.

Coordinates of points in Cantor sets. In the applications of this section, the com-pact set X mentioned below will in general be a solid torus.

Let Gi , 1 ≤ i ≤ M , be finite index sets and let Si = {Sg : Rn→ Rn

: g ∈ Gi }

be a set of similarities having the same coefficient λi of similitude. Let S = ∪Si .Additionally, suppose that there exists a compact set X ⊂ Rn such that

(1) Sg(X)⊂ X for each g ∈ Gi and

(2) the sets Sg(X) are pairwise disjoint, g ∈ Gi .

Let T = (n1, n2, . . . ) be a fixed sequence where each ni is in {1, . . . ,M}. LetGk

= Gn1 × Gn2 × · · · × Gnk , G∞

k =∏

∞

i=k Gni and G∞=∏

∞

i=1 Gni . For eachmultiindex γ = (g1, g2, . . . gk) ∈ Gk , write

Sγ = Sg1 ◦ Sg2 ◦ . . . ◦ Sgk and Xγ = Sγ (X).

In particular, Xg = Sg(X) for g ∈ G.The number of components of a multiindex γ is called the depth of γ . So

depth γ = k if γ ∈ Gk .

Let Xk =

⋃depth γ=k

Xγ .

It is well-known [Hutchinson 1981] that the intersection of the sequence of setsX ⊃ X1 ⊃ X2 ⊃ · · · is a Cantor set. Denote this set by |(S,T)|. This Cantor set isself-similar if T is repeating.

For an infinite multiindex γ = (g1, g2, g3, . . .) ∈ G∞ define

γ k= (g1, g2, . . . gk) and Xγ =

∞⋂k=1

Xγ k .

LIPSCHITZ HOMOGENEOUS WILD CANTOR SETS 289

Obviously, each Xγ is a singleton, consisting of a point from the Cantor set |(S,T)|and for each point from |(S,T)| there exists exactly one such multiindex γ . Thecomponents of γ are called the coordinates of the corresponding point from theCantor set |(S,T)|.

Finally define a juxtaposition of multiindices as follows. If δ = (d1, d2, . . . dk)

is a finite multiindex and γ = (g1, g2, . . .) is finite or infinite then let

δγ = (d1, d2, . . . dk, g1, g2, . . .).

In the special case when depth γ = 1, we have γ = g1 and δg1 = (d1, d2, . . . dk, g1).

Antoine Cantor sets. We give a brief summary of results from [Sher 1968] and[Wright 1986]. Sher’s results are needed in our proof of our main theorem, whileWright’s are used in our observation about Antoine graphs associated with theCantor sets we construct.

An Antoine Cantor set C in R3 is a Cantor set meeting the following conditions.

(1) C has a defining sequence M1,M2, . . ., each Mi consisting of the union of afinite number of pairwise disjoint standard unknotted solid tori in R3 and M1

consisting of a single solid torus.

(2) The tori in Mi , for i ≥ 2, can be listed in a sequence Ti,1, Ti,2, . . . , Ti,n(i) sothat T j and Tk are of simple linking type if j −k = ±1 mod n and do not linkif j − k 6= ±1 mod n.

(3) The linked chain of tori Ti,1, Ti,2, . . . , Ti,n(i) have winding number greaterthan 0 in the torus at the previous stage that contains them.

If, in condition (3), the winding number is required to be 1, and if each n(i)≥ 4,we call the resulting Cantor set a simple Antoine Cantor set. Most Antoine Cantorsets in the literature, including the original one [Antoine 1920] are simple.

Sher [1968] showed that if two Antoine Cantor sets C1 and C2 with definingsequences M1,M2, . . . and N1, N2, . . . are equivalently embedded in R3, then thereis a homeomorphism h of R3 to itself such that for each i , h takes the tori in Mi

homeomorphically onto the tori in Ni . As a consequence, if it can be shown thatfor some i , no such homeomorphism exists, the two Cantor sets are inequivalentlyembedded. This is the result we will need to construct the uncountably manyinequivalently embedded Cantor sets.

Wright [1986] associates an Antoine graph 0(C) with a Antoine Cantor C withdefining sequence M1,M2, . . . . The graph 0 is a countable union of nested sub-graphs 00 ⊂ 01 ⊂ 02 ⊂ · · · .The subgraph 00 is a single vertex. For each vertexv of 0i − 0i−1 , there is a polygonal simple closed curve with at least 4 verticesP(v) contained in 0i+1 −0i so that if v and w are distinct vertices of 0i −0i−1,then P(v) and P(w) are disjoint. 0i+1 consists of 0i , the union of the P(v) for v


in 0i , and the union of edges running from v to the vertices of P(v). The verticesof 0i −0i−1 correspond to the components of Mi .

If v is a vertex of 0i −0i−1 corresponding to a component C of Mi , then P(v)contains precisely the vertices corresponding to the components of Mi+1 containedin C , and two such vertices are joined by an edge if and only if the correspondingcomponents of Mi+1 are linked. Here is a diagram of an Antoine graph similar tothat in [Wright 1986, p. 252]:

Wright shows that if C1 and C2 are simple Antoine Cantor sets with differentAntoine graphs 0(C1) and 0(C2), the Cantor sets are inequivalently embedded.

In our construction, all of the Cantor sets constructed have the same Antoinegraph, but are inequivalently embedded.

3. Constructing Lipschitz homogeneous Cantor sets

Let Gi , 1 ≤ i ≤ M , Si = {Sg : Rn→ Rn

; g ∈ Gi }, X , Xg and T = (n1, n2, . . . )

be as above. The setting to keep in mind when reading Theorem 1 below is thatof a simple Antoine Cantor set defined by tori where each stage m torus has |Gnm |

stage m + 1 tori in its interior. For Theorem 1, also assume that each Gi is a finitecyclic group, with the group operation written additively.

Theorem 1. For each i , 1 ≤ i ≤ M , suppose that fi : Rn→ Rn is a Lipschitz

homeomorphism and that

(i) fi |Rn−X = id,

(ii) for each g ∈ Gi , we have fi (Xg)= Xg+1 and the diagram

X

Xgfi -

�

S g

Xg+1

Sg+1-

commutes.

Then |(S,T)| is Lipschitz homogeneous in Rn .


Proof. The approach to the proof is similar to that used in [Malesic and Repovs1999, Lemma 1]. The main modification needed is to take into account the presenceof more than one finite index set. Fix an arbitrary pair of points a and b in |(S,T)|.We will construct a homeomorphism

h : (Rn, |(S,T)|, a)→ (Rn, |(S,T)|, b)

and prove that both h and h−1 are Lipschitz. Let α = (a1, . . . , ak, . . . ) and β =

(b1, . . . , bk, . . . ) ∈ G∞ be the coordinates of a and b. For an arbitrary γ =

(g1, . . . , gk) ∈ G k define the homeomorphism

fγ = Sγ ◦ fnk+1 ◦ S−1γ : Rn

→ Rn .

Set

r1 = f b1−a1n1

, r2 = f b2−a2b1

, r3 = f b3−a3(b1,b2)

, . . . , rk+1 = f bk+1−ak+1βk , . . .

`i = r−1i ,

hk = rk ◦ rk−1 ◦ · · · ◦ r2 ◦ r1.

In addition, for notational convenience, let

r−1 = r0 = id|Rn .

It follows by Lemma 2(iv) below that the sequences of homeomorphisms h1, h2, . . .

and h−11 , h−1

2 , . . . converge pointwise at all points different from the point a andb, respectively. The convergence of the sequences at the point a and at the pointb follows from Lemma 2(ii). Denote the limits of the sequences by h : Rn

−→ Rn

and h : Rn−→ Rn , respectively. It also follows by Lemma 2 that h(a) = b, that

h(|(S,T)|)= |(S,T)|, and that h ◦ h = h ◦ h =idRn . It follows from Lemma 3 thath and h are Lipschitz. Thus Theorem 1 is proved. �

Lemmas needed for Theorem 1.

Lemma 1. The homeomorphism fγ is Lipschitz with Lipschitz constant equal tothat of fnk+1 . Moreover:

(i) fγ |Rn−Xγ = id.

(ii) For arbitrary g ∈ Gnk+1 , we have fγ (Xγ g)= Xγ (g+1), the diagram

X

Xγ gfγ -

�

S γ g

Xγ (g+1)

Sγ(g+1)-

commutes, and fγ |Xγ g is an isometry.


(iii) For arbitrary (gk+1, gk+2, . . .) ∈ G∞

k+1,

fγ (X(γ,gk+1,gk+2,...))= fγ (X(γ,1+gk+1,gk+2,...)).

Proof. This follows in a manner similar to [Malesic and Repovs 1999, Lemma 2].Part (i) follows directly from condition (i) of Theorem 1. Part (ii) follows fromcondition (ii) of the same theorem. Finally, (ii) implies (iii). �

Lemma 2. The homeomorphisms hk exhibit the following properties:

(i) h−1k = `1 ◦ `2 ◦ · · · ◦ `k−1 ◦ `k .

(ii) hk(Xαk )= Xβk and hk(Xαkγ )= Xβkγ for arbitrary multiindex γ .

(iii) the restriction hk |Xαk ak+1

: Xα kak+1 → Xβ kak+1 is an isometry.

(iv) hk |Rn−Xαk

= hk+1|Rn−Xαk

= hk+2|Rn−Xαk

= · · · .

Proof. This follows in a manner similar to [Malesic and Repovs 1999, Lemma 3].Property (i) can be proved directly by examining the construction of hk . Property(ii) follows from Lemma 1(ii)–(iii). Property (iii) holds since fγ |Xγ gk+1

is an isom-etry. Property (iv) holds because of Lemma 1(i). �

Lemma 3. hk and hk−1 are Lipschitz with equal Lipschitz constants for all valuesof k.

Proof. This requires the most modification of [Malesic and Repovs 1999] as mul-tiple similarities with different constants of similarity are involved.We fix the sequence α = (a1, a2, . . .) of coordinates of the point a ∈ |S| and intro-duce the notion of depth of a point x ∈ Rn:

dep x = j if x ∈ Xα j − Xα j+1 .

Additionally, let

dep x = 0 if x ∈ X − Xa1 and dep x = −1 if x ∈ Rn− X .

In the case x ∈ Xα j for all j ∈ N (i.e. x = a) let dep x = ∞.

For arbitrary distinct points x, y ∈ Rn we now estimate the expression hk(x)−hk(y). We may assume that dep x ≤ dep y. As x and y are distinct, case dep y =∞

and dep x = ∞ is not possible.

Case 1 Let the Lipschitz constant of the homeomorphism fi be denoted by λi . Letλ= max{λi ; 1 ≤ i ≤ M} and T = max{|Gi |; 1 ≤ i ≤ M}, where |Gi | denotes thenumber of elements of Gi . Hence the Lipschitz constants of the homeomorphismsr1, r2, . . . , `1, `2, . . . do not exceed the number3= λT . Let dep y −dep x ≤ 1, i.e.

dep x ∈ { j, j + 1}, dep y = j + 1


for some j ∈ N∪{−1, 0}. By Lemma 2, (iii) and (iv), and because of the construc-tion of hk ,

|hk(x)− hk(y)| = |r j+1 ◦ r j (x)− r j+1 ◦ r j (y)| ≤32|x − y|.

Case 2 Now let dep y − dep x ≥ 2. First let the degrees be nonnegative, i.e.

dep x = j ≥ 0 and dep y ≥ j + 2

for some j ∈ N. (It may be dep y = ∞ as well.) Then

x ∈ Xα j − Xα j+1, y ∈ Xα j+2 .

For arbitrary disjoint compact sets C1,C2 ⊂ Rn , set

dmin(C1,C2)= min{|x − y|; x ∈ C1, y ∈ C2},

dmax(C1,C2)= max{|x − y|; x ∈ C1, y ∈ C2}.

The sets X − X1 and X2 are compact and disjoint; hence the numbers

dX = dmin(X − X1, X2) and DX = dmax(X − X1, X2)

exist. Since the similarity Sαk maps the triple (X, Xa1, X(a1,a2)) onto the triple(Xαk , Xαka1, Xαk(a1,a2)), for each k ∈ N, we have

dmax(Xαk − Xαka1, Xαk(a1,a2))

dmin(Xαk − Xαka1, Xαk(a1,a2))≤

DX

dX.

Hence |hk(x)− hk(y)| ≤ (DX/dX ) |x − y|.

Finally, let dep x = −1 and dep y ≥ 1, i.e., x ∈ Rn− X and y ∈ X1. Then

hk(x)= x and

|hk(x)− hk(y)||x − y|

≤|x − y| + |y − hk(y)|

|x − y|≤ 1 +

diam X1

m,

where m = inf{|x − y|; x ∈ Rn− X , y ∈ X1} (it is easy to show that m > 0). To

conclude, set

L = max{32,

DX

dX, 1 +

diam X1

m

}.

Then |hk(x)− hk(y)| ≤ L|x − y| for an arbitrary k ∈ N and x, y ∈ Rn .The estimate |h−1

k (x)− h−1k (y)| ≤ L|x − y| can be proved analogously, using

Lemma 2(i). �


4. Main result

Theorem 2. There exist uncountably many topologically distinct Lipschitz homo-geneous wild Cantor sets in R3. In fact, these Cantor sets can all be constructed assimple Antoine Cantor sets with the same number of components of stage n insideeach component of stage n − 1 and thus with the same Antoine graphs.

Proof. Use Theorem 1 with M = 2, G1 = Z60 and G2 = Z60. Let T = (n1, n2, . . . )

be a fixed sequence of 1’s and 2’s. For each such sequence, construct a Lipschitzhomogeneous Antoine Cantor set as in Theorem 1.

For G1, let the similarities Sg, g ∈ G1 be constructed so as to take the outer torusin Figure 1 to the smaller tori in the same figure. Each smaller torus in the chainis obtained from the previous one by rotating the large torus by 2π/60 radians andthen by rotating the small tori by π/2 radians. The homeomorphism f1 needed inTheorem 1 is constructed in a manner similar to that constructed in the example in[Malesic and Repovs 1999].

For G2, let the similarities Sg, g ∈ G2 be constructed so as to take the outertorus in Figure 2 to the smaller tori in Figure 2. Each smaller torus in the chain isobtained from the previous one by rotating the large torus by 2π/60 radians andthen by rotating the small tori by π/4 radians. The homeomorphism f2 needed inTheorem 1 is constructed in a manner similar to that constructed in the example in[Malesic and Repovs 1999]. The resulting Cantor set is Lipschitz homogeneouslyembedded by Theorem 1.

Note that the Antoine graphs associated with any two Cantor set constructed inthis way are the same.

Figure 1. Left: A torus with 60 smaller similar tori linked in asimple chain inside, each of which is rotated by π/2 radians fromthe previous one. Right: an enlargement of five of the smaller tori.


Let C1 be the Cantor set constructed as in Theorem 1 for one sequence of 1’sand 2’s and let C2 be the Cantor set constructed using a different sequence. Weneed to show that these two Cantor sets are topologically inequivalently embedded.For this, using the result of Sher mentioned on page 289, it suffices to show thatthere is no homeomorphism of R3 to itself taking the large torus in Figure 1 to thelarge torus in Figure 2, and taking the chain of smaller tori in Figure 1 to the chainof smaller tori in Figure 2. If there were such a homeomorphism, the link formedby the centerlines of the small tori in Figure 1 would be topologically equivalentthe link formed by the centerlines of the small tori in Figure 2. The next lemmashows that this is not the case.

There are uncountably many sequences of 1’s and 2’s. The argument aboveshows that each such sequence leads to a topologically distinct Lipschitz homoge-neous wild Cantor set. This completes the proof of the theorem. �

Lemma 4. The links formed by the center lines of the smaller tori in Figures 1 and2 are inequivalent.

Proof. The link formed by the centerlines of the chain of smaller tori in Figure 2 isseen to be topologically embedded in R3 as follows. Starting at a fixed centerlineand proceeding around the chain, the centerline can be twisted by a homeomor-phism of R3 fixed outside of the large torus in such a way that all but one of thecenterlines are embedded in the same manner as the link formed by the centerlinesof the smaller tori in Figure 1. The remaining centerline in Figure 2 can be ob-tained from the corresponding centerline in Figure 1 by the following modification.One of the centerlines in Figure 1 is given 30 half twists before linking with thecenterlines on either side. A computation of the version of the Jones polynomialintroduced in [Kauffman 1988] shows that these links are topologically distinct,

Figure 2. A similar chain, with each small torus rotated π/4 rel-ative to the previous one.


and thus the Cantor sets are topologically distinct. In fact, all that is needed is toshow that the span of the Kauffman polynomial (the highest power appearing inthe polynomial minus the lowest power) is different in each case.

For details of a computation of the Kauffman polynomial in this setting, see[Garity ≥ 2006]. For completeness, we outline the computation. We refer thereader to [Kauffman 1988] for details on the Kauffman bracket and Kauffmanpolynomial. Let L represent an oriented link and |L| represent a particular diagramfor this link. The writhe of L in the diagram |L| is denoted by ω(|L|). Let L+, L−,

and L0 represented oriented link diagrams identical to |L| except at one crossing,where L+ represents this crossing with positive crossing number, L− with negativecrossing number and L0 with the crossing split and the orientation preserved. Weuse X (L) to denote the Kauffman polynomial of L .

Open Chains. Let Cn be a simple chain with n links. An easy induction, or themultiplicativity of the Kauffman bracket relative to connected sums, shows that〈Cn〉 = (−A4

− A−4)n−1, where 〈〉 is the Kauffman bracket. If we orient the linksso the linking numbers alternate signs as in the superscripts below, we have

X (C + −···+−

n )= X (C − +···− +

n )= (−A4− A−4)n−1 for n odd,

X (C + −···− +

n )= (−A)−6(−A4− A−4)n−1 for n even,

X (C − + ··· + −

n )= (−A)6(−A4− A−4)n−1 for n even.

In each case, the span of the polynomial is 8n−1 and the maximum and minimumexponent in the polynomial can be read off. The maximum exponent is 4(n −1) ifn is odd and 4(n − 1)± 6 if n is even. The minimum exponent is −4(n − 1) if nis odd and −4(n − 1)± 6 if n is even.

Closed Chains. Now take a closed chain LC2n with 2n components and with notwists, as in Figure 1. Orient it so the linking numbers are alternately positive andnegative. Consider three consecutive links and modify them as follows:

L1 L2

L21 L22


We now compute using the relation A4 X (+)= A−4 X (−)+ (A−2− A2)X (⇒)

and noting that the link labeled L1 above is C+···+

2n , the link L21 is C−···+

2n−1 and thelink L22 is LC2n−2:

X (LC2n)= A8 X (L1)+ A4(A2− A−2)X (L2)

= A8 X (C+···+

2n )+ A4(A2−A−2)

(A−8 X (L21)+ A−4(A−2

−A2)X (L22))

= A8 X (C+···+

2n )

+ A4(A2− A−2)

(A−8 X (C−···+

2n−1 )+ A−4(A−2− A2)X (LC2n−2)

)= · · ·

= (−A4− A−4)2n−2(−A6

− A−6)− (A−2− A2)2 X (LC2n−2).

Thus

X (LC2n)= (−A4− A−4)2n−2(−A6

− A−6)− (A−2− A2)2 X (LC2n−2)

This sets up a recursion relation that can be solved for X (LC2n). The startingcondition is that X (LC2·1)= (−A2

−A−2) has maximum exponent 2 and minimumexponent −2, and hence span 4.

The maximum exponent of X (LC2n) is

max{4(2n − 2)+ 6, 4 + maximum exponent(X (LC2n−2))}

and the minimum exponent of X (LC2n) is

min{−4(2n − 2)− 6,minimum exponent((LC2n−2))− 4}.

An easy induction now shows that span (X (LC2·n))=16(n − 1)+ 12 for n ≥ 2.

Closed chains with twists. Take the case of a linked chain forming a loop with 2ncomponents, with k positive half twists, where k is even, in one of the links, LCk

2n .Orient as in the previous case. By considering the diagram

one can make the following computations.

X (LCk2n)= A−8 X (LCk−2

2n )+ A−4(A−2− A2)X (C2n+1),

X (LCk2n)= A−8 X (LCk−2

2n )+ A−4(A−2− A2)(−A−4

− A4)2n.

The starting point here is when k = 0, LC02n = LC2n


The maximum exponent is max{max. exp.(LC2k−22n )− 8, 8n − 2}.

The minimum exponent is min{min. exp.(LC2k−22n )− 8,−8n − 6}.

An easy induction now shows that span(X (LC2k2n))=16n−4+4k, distinguishing

topologically all the chains with 2n links and different numbers of even twists. Thiscompletes the proof of the lemma. �

5. Other results and questions

Using techniques similar to those used in the proof of Theorem 1, we can prove thefollowing result. Note that in this case, we assume that G is of the form Z p × Zq

for some positive integers p and q.

Theorem 3. For each i , 1 ≤ i ≤ 2, suppose that fi : Rn→ Rn is a Lipschitz

homeomorphism and that:

(i) fi |Rn−X = id;

(ii) f1(X(a,b))= X(a+1,b) for (a, b) ∈ G;

(iii) f2(X(a,b))= X(a,b+1) for (a, b) ∈ G and the following diagrams commute:

X X

X(a,b)f1 -

�S (a,b)

X(a+1,b)

S(a+1,b)-

X(a,b)f2 -

�S (a,b)

X(a,b+1)

S(a,b

+1)-

Then |(S,T)| is Lipschitz homogeneous in Rn .

The construction suggested by the theorem is similar to the Blankinship con-struction [1951] for wild Cantor sets in R4.

Question. Can this theorem be used to show that a Lipschitz homogeneous wildCantor set in R4 exists? This would require a more careful Blankinship-type con-struction, in which the successive stages in the construction were self-similar tothe original stage.

Acknowledgment

We thank the referee for helpful suggestions.

References

[Antoine 1920] M. L. Antoine, “Sur la possibilité d’étendre l’homéomorphie de deux figures à leurvoisinages”, C. R. Acad. Sci. Paris 171 (1920), 661–663. JFM 47.0524.01

[Blankinship 1951] W. A. Blankinship, “Generalization of a construction of Antoine”, Ann. of Math.(2) 53 (1951), 276–297. MR 12,730c Zbl 0042.17601

[Eaton 1973] W. T. Eaton, “A generalization of the dog bone space to En”, Proc. Amer. Math. Soc.39 (1973), 379–387. MR 48 #1238 Zbl 0262.57001


[Garity ≥ 2006] D. J. Garity, “Inequivalent Antoine Cantor sets with the same Antoine tree”, inProceedings of the Twentieth Annual Workshop in Geometric Topology (Park City, UT, 2003). Toappear.

[Hutchinson 1981] J. E. Hutchinson, “Fractals and self-similarity”, Indiana Univ. Math. J. 30:5(1981), 713–747. MR 82h:49026 Zbl 0598.28011

[Kauffman 1988] L. H. Kauffman, “New invariants in the theory of knots”, Amer. Math. Monthly95:3 (1988), 195–242. MR 89d:57005 Zbl 0657.57001

[Malešic and Repovš 1999] J. Malešic and D. Repovš, “On characterization of Lipschitz manifolds”,pp. 265–277 in New developments in differential geometry (Budapest, 1996), edited by J. Szenthe,Kluwer, Dordrecht, 1999. MR 99j:57042 Zbl 0939.57023

[Repovš et al. 1996] D. Repovš, A. B. Skopenkov, and E. V. Šcepin, “C1-homogeneous compacta inRn are C1-submanifolds of Rn”, Proc. Amer. Math. Soc. 124:4 (1996), 1219–1226. MR 1301046(97f:58008) Zbl 0863.53004

[Sher 1968] R. B. Sher, “Concerning wild Cantor sets in E3”, Proc. Amer. Math. Soc. 19 (1968),1195–1200. MR 38 #2755 Zbl 0165.57202

[Wright 1986] D. G. Wright, “Rigid sets in En”, Pacific J. Math. 121:1 (1986), 245–256. MR 87b:57011 Zbl 0697.12018 Zbl 0586.57009


DENNIS GARITY

MATHEMATICS DEPARTMENT

OREGON STATE UNIVERSITY

CORVALLIS, OR 97331UNITED STATES

[email protected]://www.math.oregonstate.edu/˜garity

DUSAN REPOVS

INSTITUTE OF MATHEMATICS, PHYSICS AND MECHANICS

UNIVERSITY OF LJUBLJANA

JADRANSKA 19, P.O.BOX 2964LJUBLJANA

SLOVENIA

[email protected]

MATJAZ ZELJKO

INSTITUTE OF MATHEMATICS, PHYSICS AND MECHANICS

UNIVERSITY OF LJUBLJANA

JADRANSKA 19, P.O.BOX 2964LJUBLJANA

SLOVENIA

[email protected]


NOETHER’S PROBLEM FOR DIHEDRAL 2-GROUPS II

MING-CHANG KANG

Let K be any field and G be a finite group. Let G act on the rational functionfield K (xg : g ∈ G) by K -automorphisms defined by g · xh = xgh for anyg, h ∈ G. Denote by K (G) the fixed field K (xg : g ∈ G)G . Noether’s problemasks whether K (G) is rational (= purely transcendental) over K . A result ofSerre shows that Q(G) is not rational when G is the generalized quaterniongroup of order 16. We shall prove that K (G) is rational over K if G is anynonabelian group of order 16 except when G is the generalized quaterniongroup of order 16. When G is the generalized quaternion group of order 16and K (ζ8) is a cyclic extension of K , then K (G) is also rational over K .

1. Introduction

Let K be any field and G be a finite group. Let G act on the rational function fieldK (xg : g ∈ G) by K -automorphisms such that g ·xh = xgh for any g, h ∈ G. Denoteby K (G) the fixed subfield

K (xg : g ∈ G)G = { f ∈ K (xg : g ∈ G) : σ · f = f for any σ ∈ G}.

Noether’s problem asks whether K (G) is rational (that is, purely transcendental)over K .

Noether’s problem for finite abelian groups has been studied by Swan, Endoand Miyata, Voskresenskii, Lenstra, Colliot-Thelene and Sansuc, among others;see [Swan 1983] and the references therein. But our knowledge about Noether’sproblem for nonabelian groups is rather incomplete. It is known that K (G) isrational if G is a transitive solvable subgroup of the symmetric group Sp when p =

3, 5, 7, 11 [Furtwangler 1925], the quaternion group of order 8 [Grbner 1934], thealternating group A5 [Maeda 1989; Kervaire and Vust 1989], P SL2(7), P Sp4(3)(such that the base fields K contain suitable quadratic fields of Q) [Kemper 1996]or finite reflection groups [Kemper and Malle 1999]. Noether’s problem for meta-abelian groups and dihedral groups is discussed in [Haeuslein 1971; Hajja 1983;

MSC2000: 12F12, 13A50, 11R32, 14E08.Keywords: rationality, Noether’s problem, generic Galois extensions, generic polynomials, groups

of order 16.Partially supported by the National Science Council, Republic of China.

301

302 MING-CHANG KANG

Kang 2004]. One striking result is Saltman’s Theorem [1984], which shows thatC(G) is never rational for a p-group of order p9. (See [Bogomolov 1987] for p-groups of smaller orders.) If K is a field containing enough roots of unity, thenK (G) is rational for any nonabelian group of order p3 or p4 [Chu and Kang 2001].A result of Serre [1995, 3.5] (see also [Garibaldi et al. 2003, Theorem 33.26 andExample 33.27, pp. 89–90]) shows that Q(G) is not rational if G is the generalizedquaternion group of order 16 (see Theorem 1.3 for the definition of this group); infact, it is shown that, if G is a finite group whose 2-Sylow subgroup is isomorphicto the generalized quaternion group, then Q(G) is not rational [Garibaldi et al.2003, Theorem 34.7, p. 92]. Thus it would be interesting to investigate for whichfields K and 2-groups G the field K (G) will be rational, at least for groups ofsmall order. It turns out that, if G is a nonabelian group of order 8 or 16, Serre’scounterexample is the only exceptional case. See Theorem 1.3.

One motivation to study Noether’s problem arises from the inverse Galois prob-lem, in particular, the existence of a generic polynomial for G-extensions over K(equivalently, the existence of a generic Galois G-extension over K ). If K is aninfinite field and K (G) is rational over K , there exists a generic polynomial for G-extensions over K [Saltman 1982, Theorem 5.1; DeMeyer and McKenzie 2003].(See also [Hashimoto and Miyake 1999] for the case of dihedral extensions.) Formost p-groups G, it is still unknown whether a generic Galois G-extension overK exists [Saltman 1982]. We just mention some relevant results:

Theorem 1.1. Let K be any infinite field.

(1) [Black 1999] There exists a generic Galois G-extension over K if G = D4 orD8, where D4 and D8 are the dihedral groups of order 8 and 16.

(2) [Ledet 2000; 2001] There exists a generic polynomial for G-extensions overK , if G is

(i) a nonabelian group of order 8, or(ii) a nonabelian group of order 16 defined by

G = 〈σ, τ : σ 8= τ 2

= 1, τ−1στ = σ a〉 with a = 3, 5, 7, or

G = 〈σ, τ, λ : σ 4= τ 2

= λ2= 1, τ−1στ = λ−1σλ= σ, λ−1τλ= σ 2τ 〉.

Theorem 1.2 [Chu et al. 2004]. For any field K , K (G) is rational over K providedthat G is

(i) a nonabelian group of order 8, or

(ii) a nonabelian group of order 16 defined by

G = 〈σ, τ : σ 8= τ 2

= 1, τ−1στ = σ a〉 with a = 3, 5 or 7.

NOETHER’S PROBLEM FOR DIHEDRAL 2-GROUPS II 303

What we will prove in this article completes our knowledge of Noether’s prob-lem for groups of order 16:

Theorem 1.3. For any field K , K (G) is rational over K , if G is any nonabeliangroup of order 16 except possibly the generalized quaternion group defined by

G = 〈σ, τ : σ 8= τ 4

= 1, σ 4= τ 2, τ−1στ = σ−1

〉.

For this “exceptional” group G, if K (ζ8) is cyclic over K where ζ8 is a primitive8-th root of unity (in case char K 6= 2), then K (G) is rational also.

As mentioned before we cannot improve the “exceptional” group G in Theorem1.3 if K = Q because of Serre’s Theorem. We will remark that the novelty ofTheorem 1.3. is that no “unnecessary” restriction on the field is assumed; it isknown that K (G) is always rational provided that G is any nonabelian p-group oforder p3 or p4 with exponent pe and K is a field containing a primitive pe-th rootof unity [Chu and Kang 2001, Theorem 1.6].

As an application of the above theorem we obtain the following theorem, thanksto [Saltman 1982, Theorem 5.1].

Theorem 1.4. For any infinite field K , a generic Galois G-extension over K existsfor any nonabelian group G of order 16, except possibly the generalized quaterniongroup of order 16 defined in Theorem 1.3. If G is the generalized quaternion groupof order 16 and K (ζ8) is cyclic over K (in case char K 6= 2), then a generic GaloisG-extension over K exists also.

We shall organize this paper as follows. We will recall some preliminaries inSection 2. Theorem 1.3 will be proved in Section 3. We will remark that, sincethe proof of this theorem is constructive, a transcendental basis of K (G) can beexhibited explicitly. Thus a generic polynomial for G-extensions over K can befound by applying [Kemper and Malle 1999, Proposition 3.1]. Since Noether’sproblem for finite abelian groups was completely solved by Lenstra [1974], wewill concentrate on nonabelian groups.

Notations and terminologies. A field extension L over K is rational if L is purelytranscendental over K ; L is called stably rational over K if there exist elementsy1, . . . , yN which are algebraically independent over L such that L(y1, . . . , yN ) isrational over K . ζn will denote a primitive n-th root of unity in some extensionfield of the field K when char K = 0 or char K = p > 0 with p - n. Finally, recallthe definition K (G) at the beginning of this section: K (G) = K (xg : g ∈ G)G .The representation space of the regular representation of G over K is denoted byW =

⊕g∈G K · x(g) where G acts on W by g · x(h)= x(gh) for any g, h ∈ G.

304 MING-CHANG KANG

2. Generalities

We recall a variant of Hilbert’s Theorem 90 that has been used by many peopleunder different guises.

Theorem 2.1 [Hajja and Kang 1995, Theorem 1]. Let L be a field and G a finitegroup acting on L(x1, . . . , xm), the rational function field of m variables over L .Suppose that

(i) for any σ ∈ G, σ(L)⊂ L;

(ii) the restriction of the action of G to L is faithful;

(iii) for any σ ∈ G, σ(x1)

.

.

.

σ (xm)

= A(σ )

x1

.

.

.

xm

+ B(σ )

where A(σ ) ∈ GLm(L) and B(σ ) is an m × 1 matrix over L .

Then there exist z1, . . . , zm ∈ L(x1, . . . , xm) such that

L(x1, . . . , xm)G

= LG(z1, . . . , zm)

and σ(zi )= zi for any σ ∈ G and any 1 ≤ i ≤ m.

Theorem 2.2 [Hajja and Kang 1994, Lemma (2.7)]. Let K be any field, a, b ∈

K \ {0} and σ : K (x, y)→ K (x, y) the K -automorphism defined by σ(x) = a/x ,σ(y)= b/y. Then K (x, y)〈σ 〉

= K (u, v) where

u =

x −ax

xy −abxy

, v =

y −by

xy −abxy

.

Moreover, x + (a/x) = (−bu2+ av2

+ 1)/v, y + (b/y) = (bu2− av2

+ 1)/u,xy + (ab/(xy))= (−bu2

− av2+ 1)/(uv).

Theorem 2.3 [Kuniyoshi 1955; Miyata 1971]. Let K be a field with char K =

p> 0 and G be a p-group. Then K (V )G is rational over K for any representationρ : G → GL(V ) where V is a finite-dimensional vector space over K .

Proof. Since char K = p > 0 and |G| = pm , any representation of G can betriangulated. Apply [Hajja and Kang 1994, Theorem (2.2)]. �


3. Proof of Theorem 1.3

Without loss of generality we will assume that K is any field with char K 6= 2throughout this section, because Theorem 2.3 will take care of the case char K = 2.

Here is a list of nonabelian groups of order 16, which can be found in [Huppert1967, p. 349] or in [Chu and Kang 2001, Theorem 3.4]:

(I) 〈σ, τ : σ 8= τ 2

= 1, τ−1στ = σ−1〉,

(II) 〈σ, τ : σ 8= τ 4

= 1, σ 4= τ 2, τ−1στ = σ−1

〉,

(III) 〈σ, τ : σ 8= τ 2

= 1, τ−1στ = σ 5〉,

(IV) 〈σ, τ : σ 8= τ 2

= 1, τ−1στ = σ 3〉,

(V) 〈σ, τ, λ : σ 4= τ 2

= λ2= 1, τ−1στ = λ−1σλ= σ, λ−1τλ= σ 2τ 〉,

(VI) 〈σ, τ, λ : σ 4= τ 2

= λ2= 1, τ−1στ = σ−1, λ−1σλ= σ, λ−1τλ= τ 〉,

(VII) 〈σ, τ, λ:σ 4=τ 4

=λ2=1, σ 2

=τ 2, τ−1στ=σ−1, λ−1σλ=σ, λ−1τλ=τ 〉,

(VIII) 〈σ, τ : σ 4= τ 4

= 1, τ−1στ = σ−1〉,

(IX) 〈σ, τ, λ : σ 4= τ 2

= λ2= 1, τ−1στ = σ, λ−1σλ= στ, λ−1τλ= τ 〉.

Because we have solved the rationality problem for the groups (I), (III), (IV) in[Chu et al. 2004], we will consider the remaining six groups in this article.

Case 1. The group (V): G = 〈σ, τ, λ : σ 4= τ 2

= λ2= 1, τ−1στ = λ−1σλ =

σ, λ−1τλ= σ 2τ 〉.If

√−1 ∈ K , then K (G) is rational by [Chu and Kang 2001, Theorem 1.6].

Hence we shall assume that√

−1 /∈ K from now on.Let W =

⊕g∈K K ·x(g) be the representation space of the regular representation

of G. Define

x1 = x(1)+ x(τ )− x(σ 2)− x(σ 2τ),

x2 = σ · x1, x3 = λ · x1, x4 = λσ · x1.

Then we find that

σ : x1 7→ x2 7→ −x1, x3 7→ x4 7→ −x3,

τ : x1 7→ x1, x2 7→ x2, x3 7→ −x3, x4 7→ −x4,

λ : x1 ↔ x3, x2 ↔ x4.

Moreover,⊕

1≤i≤4 K · xi is a faithful G-subspace of W . Thus K (G) is rationalif K (x1, . . . , x4)

G is rational by Theorem 2.1.Let Gal(K

√−1/K ) = 〈ρ〉 and ρ :

√−1 7→ −

√−1. We extend the actions of

σ , τ , λ, ρ to K (√

−1)(x1, . . . , x4) by requiring σ(√

−1)= τ(√

−1)= λ(√

−1)=

306 MING-CHANG KANG

√−1, ρ(xi )= xi for 1 ≤ i ≤ 4. Then

K (x1, . . . , x4)〈σ,τ,λ〉

= {K (√

−1)(x1, . . . , x4)〈ρ〉

}〈σ,τ,λ〉

= K (√

−1)(x1, . . . , x4)〈σ,τ,λ,ρ〉.

Define

y1 =√

−1x1 + x2, y2 = −√

−1x1 + x2,

y3 =√

−1x3 + x4, y4 = −√

−1x3 + x4.

Then we get

σ : y1 7→√

−1 y1, y2 7→ −√

−1 y2, y3 7→√

−1 y3, y4 7→ −√

−1 y4,

τ : y1 7→ y1, y2 7→ y2, y3 7→ −y3, y4 7→ −y4,

λ : y1 ↔ y3, y2 ↔ y4,

ρ : y1 ↔ y2, y3 ↔ y4.

Definez1 = y1 y2, z2 = y3 y4, z3 = y3/y1, z4 = y4

1 .

Then K (√

−1)(y1, . . . , y4)<σ>

= K (√

−1)(z1, . . . , z4); moreover,

τ : z1 7→ z1, z2 7→ z2, z3 7→ −z3, z4 7→ z4,

λ : z1 ↔ z2, z3 7→ 1/z3, z4 7→ z43z4,

ρ : z1 7→ z1, z2 7→ z2, z3 7→ z2/(z1z3), z4 7→ z41/z4.

Thus K (√

−1)(z1, . . . , z4)〈τ 〉

= K (√

−1)(z1, z2, z23, z4).

Define

u1 = z1z2, u2 = z23z4/(z1z2), x = z1, y = z1z2

3/z2.

Then we find that

λ : u1 7→ u1, u2 7→ u2, x 7→ a/x, y 7→ b/y,

ρ : u1 7→ u1, u2 7→ 1/u2, x 7→ x, y 7→ 1/y.

where a = u1, b = 1.Define

u =

x −ax

xy −abxy

, v =

y −by

xy −abxy

.

By Theorem 2.2, K (√

−1)(u1, u2, x, y)〈λ〉 = K (√

−1)(u1, u2, u, v).


It is routine to check that

ρ : u 7→

x −ax

bxy

−ayx

, v 7→ −

y −by

bxy

−ayx

.

Define w = u/v. Then ρ(w)= −w. It is not difficult to verify that

(3–1)x −

ax

bxy

−ayx

=u

bu2 − av2 .

In fact, using Theorem 2.2, the right-hand side of (3–1) is equal to (y + (b/y)−(1/u))−1. It is very easy to check that the left-hand side of (3–1) is equal to thesame quantity.

It follows that ρ(u)= u/(bu2− av2)= c/u where c = w2/(bw2

− a).Define

t = u1, s =√

−1w, q = w(1 + u2)/(1 − u2).

Then

ρ :√

−1 7→ −√

−1, t 7→ t, s 7→ s, q 7→ q, u 7→ c/u

where c = w2/(bw2− a)= s2/(s2

+ t).Define p = (s2

+ t)u/s. Then ρ(p)= A/p where A = s2+ t .

It follows that K (√

−1)(u1, u2, u, v)〈ρ〉= K (

√−1)(t, s, p, q)〈ρ〉

= K (√

−1)(t, s, p)〈ρ〉(q) = K (t, s, X, Y, q) where X = p + (A/p), Y =

√−1(p − (A/p)).

Note that a relation of X and Y is

X2+ Y 2

= 4A = 4(s2+ t).

Hence t ∈ K (s, X, Y ). It follows that K (t, s, X, Y, q)= K (s, X, Y, q) is rationalover K .

Case 2. The group (VI): G =〈σ, τ, λ :σ 4= τ 2

=λ2= 1, τ−1στ =σ−1, λ−1σλ=

σ, λ−1τλ= τ 〉.As before, let W =

⊕g∈G K · x(g) be the regular representation of G. Define

x1 = x(1)+ x(τ )− x(σ 2)− x(σ 2τ),

x2 = σ · x1, x3 = λ · x1, x4 = λσ · x1.

308 MING-CHANG KANG

Then we find that

σ : x1 7→ x2 7→ −x1, x3 7→ x4 7→ −x3,

τ : x1 7→ x1, x2 7→ −x2, x3 7→ x3, x4 7→ −x4,

λ : x1 ↔ x3, x2 ↔ x4.

As in Case 1, it suffices to consider the case√

−1 /∈ K . Let Gal(K (√

−1/K )= 〈ρ〉 with ρ(

√−1)= −

√−1.

Define yi and zi , for 1 ≤ i ≤ 4, the same way as in Case 1. We find

K (√

−1)(x1, . . . , x4)〈σ 〉

= K (√

−1)(z1, . . . , z4).

The actions of λ and ρ on z1, . . . , z4 are the same as in Case 1, while τρ(√

−1)=−

√−1, τρ(zi )= zi for 1 ≤ i ≤ 4. Thus

K (√

−1)(z1, . . . , z4)〈τ,λ,ρ〉

= K (√

−1)(z1, . . . , z4)〈τρ,λ,ρ〉

= K (z1, . . . , z4)〈λ,ρ〉.

Define

u1 = z1z2, u2 = z1, x = z3/z2, y = z23z4/(z1z2).

We find that

λ; u1 7→ u1, u2 7→ u1/u2, x 7→ a/x, y 7→ y,

ρ : u1 7→ u1, u2 7→ u2, x 7→ a/x, y 7→ b/y,

where a = 1/u1 and b = 1.Define u, v, and w= u/v the same way as in Case 1. Then K (u1, u2, x, y)〈ρ〉

=

K (u1, u2, w, u) and we find that

λ : u1 7→ u1, u2 7→ u1/u2, w 7→ −w, u 7→ c/u,

where c = w2/(bw2− a).

Define

w1 = −1/(u1w2), w2 = ww1u2, w3 = u/c.

Then K (u1, u2, w, u)= K (w1, w2, w3, w) and

λ : w1 7→ w1, w2 7→ w1/w2, w3 7→ (w1 + 1)/w3.

By Theorem 2.1, it suffices to show that K (w1, w2, w3)〈λ〉 is rational over K .

But this is easy because it is even rational over K (w1) by Theorem 2.2.

Case 3. The group (VII): G = 〈σ, τ, λ : σ 4= τ 4

= λ2= 1, σ 2

= τ 2, τ−1στ =

σ−1, λ−1σλ= σ, λ−1τλ= τ 〉.


Define a faithful G-subspace⊕

1≤i≤5 K · xi in the representation space W =⊕g∈G K · x(g) of the regular representation by

x1 = x(1)+ x(λ)− x(σ 2)− x(σ 2λ),

x2 = σ · x1, x3 = τ · x1, x4 = τσ · x1,

x5 =

∑h∈H

x(h)−∑h∈H

x(hλ)

where H is the subgroup generated by σ and τ .We find that

σ : x1 7→ x2 7→ −x1, x3 7→ −x4, x4 7→ x3, x5 7→ x5,

τ : x1 7→ x3, x2 7→ x4, x3 7→ −x1, x4 7→ −x2, x5 7→ x5,

λ : x1 7→ x1, x2 7→ x2, x3 7→ x3, x4 7→ x4, x5 7→ −x5.

Thus

K(x1, . . . , x5)〈λ〉

= K(x1, . . . , x25), K(x1, . . . , x5)

〈σ,τ,λ〉= K(x1, . . . , x4)

〈σ,τ 〉(x25).

However, the fixed field K (x1, . . . , x4)〈σ,τ 〉 is exactly the same as in the proof of

[Chu et al. 2004, Theorem 2.6]. (The subgroup H = 〈σ, τ 〉 is the quaternion groupof order 8.) Hence the result.

Case 4. The group (VIII): G = 〈σ, τ : σ 4= τ 4

= 1, τ−1στ = σ−1〉.


1≤i≤4 K · xi in W =⊕

g∈G K · x(g) by

x1 = x(1)+ x(τ )− x(σ 2)− x(σ 2τ), x2 = σ · x1,

x3 =

∑0≤i≤3

x(σ i )−∑

0≤i≤3

x(σ iτ 2), x4 = τ · x3.

Then we find

σ : x1 7→ x2 7→ −x1, x3 7→ x3, x4 7→ x4,

τ : x1 7→ x1, x2 7→ −x2, x3 7→ x4 7→ −x3.

It is clear that K (x1, . . . , x4)〈σ 2,τ 2

〉= K (y1, . . . , y4) where y1 = x1x2, y2 =

x21 , y3 = x3x4, y4 = x2

3 .Note that

σ : y1 7→ −y1, y2 7→ y21/y2, y3 7→ y3, y4 7→ y4,

τ : y1 7→ −y1, y2 7→ y2, y3 7→ −y3, y4 7→ y23/y4.

Define u1 = y2 + (y21/y2), u2 = (y1/y2)− (y2/y1), u3 = y3, u4 = y3/y4. Since

[K (u1, u2)(y1) : K (u1, u2)] ≤ 2, it follows that K (y1, . . . , y4)〈σ 〉

= K (u1, . . . , u4).

310 MING-CHANG KANG

Note that

τ : u1 7→ u1, u2 7→ −u2, u3 7→ −u3, u4 7→ −1/u4.

Hence K (u1, . . . , u4)〈τ 〉

= K (u1, u2u3, u3u4 + (u3/u4), u4 − (1/u4)) is rationalover K .

Case 5. The group (IX): G = 〈σ, τ, λ : σ 4= τ 2

= λ2= 1, τ−1στ = σ, λ−1σλ=

στ, λ−1τλ= τ 〉.


1≤i≤4 K · xi in W =⊕

g∈G K · x(g) by

x1 = x(1)− x(σ 2)− x(λ)+ x(σ 2λ),

x2 = σ · x1, x3 = τ · x1, x4 = τσ · x1.

Then we find that

σ : x1 7→ x2 7→ −x1, x3 7→ x4 7→ −x3,

τ : x1 ↔ x3, x2 ↔ x4,

λ : x1 7→ −x1, x2 7→ −x4, x3 7→ −x3, x4 7→ −x2.

Define

y1 = x1 − x3, y2 = x2 − x4, y3 = x1 + x3, y4 = x2 + x4.

It follows that

σ : y1 7→ y2 7→ −y1, y3 7→ y4 7→ −y3,

τ : y1 7→ −y1, y2 7→ −y2, y3 7→ y3, y4 7→ y4,

λ : y1 7→ −y1, y2 7→ y2, y3 7→ −y3, y4 7→ −y4.

Hence K (x1, . . . , x4)〈τ 〉

= K (y1, . . . , y4)〈τ 〉

= K (z1, . . . , z4) where z1 = y21 ,

z2 = y1 y2, z3 = y3, z4 = y4. Moreover, it can be verified that

σ : z1 7→ z22/z1, z2 7→ −z2, z3 7→ z4 7→ −z3,

λ : z1 7→ z1, z2 7→ −z2, z3 7→ −z3, z4 7→ −z4.

Define u1 = z1, u2 = z22, u3 = z2z3, u4 = z3z4. Then K (z1, . . . , z4)

〈λ〉=

K (u1, . . . , u4) and we find

σ : u1 7→ u2/u1, u2 7→ u2, u3 7→ −u2u4/u3, u4 7→ −u4.

It is easy to check that K (u1, . . . , u4)〈σ 2

〉= K (u1, u2, u2

3, u4).Define

v1 = u2, v2 = (u1 − (u2/u1))u4, x = u1, y = u23/u4.


We find that

σ : v1 7→ v1, v2 7→ v2, x 7→ v1/x, y 7→ −v1/y.

By Theorem 2.2, K (v1, v2, v3, v4)〈σ 〉 is rational over K .

Case 6. The group (II): G = 〈σ, τ : σ 8= τ 4

= 1, σ 4= τ 2, τ−1στ = σ−1

〉 andassume that K (ζ8) is cyclic over K .

If ζ8 ∈ K , then K (G) is rational over K by [Chu and Kang 2001, Theorem 1.6].Hence we shall assume that ζ8 /∈ K from now on.

Because K (ζ8) is cyclic over K , it follows that Gal(K (ζ8)/K ) = 〈ρ〉 whereρ(ζ8)= ζ a

8 with a = 3, 5 or 7.We will find a faithful G-subspace

⊕1≤i≤8 K · xi of W =

⊕g∈G K · x(g) by

x1 = x(1)− x(σ 4), xi = σ i−1x1 for 2 ≤ i ≤ 4,

x j = τσ j−5x1 for 5 ≤ j ≤ 8.

We find that

σ : x1 7→ x2 7→ x3 7→ x4 7→ −x1, x8 7→ x7 7→ x6 7→ x5 7→ −x8,

τ : x1 7→ x5, x2 7→ x6, x3 7→ x7, x4 7→ x8, x5 7→ −x1, x6 7→ −x2,

x7 7→ −x3, x8 7→ −x4.

We shall write ζ for ζ8 in the sequel and remember ρ(ζ ) = ζ a with a = 3, 5or 7. We shall extend the actions of σ, τ, ρ to K (ζ )(x1, . . . , x8) by requiringσ(ζ )= τ(ζ )= ζ and ρ(xi )= xi for 1 ≤ i ≤ 8. It follows that

K (x1, . . . , x8)〈σ,τ 〉

= {K (ζ )(x1, . . . , x8)〈ρ〉

}〈σ,τ 〉

= K (ζ )(x1, . . . , x8)〈σ,τ,ρ〉.

Define y1, y3, y5, y7, z1, z3, z5, z7 by

y1 = (σ − ζ 3)(σ − ζ 5)(σ − ζ 7) · x1, y3 = (σ − ζ )(σ − ζ 5)(σ − ζ 7) · x1

y5 = (σ − ζ )(σ − ζ 3)(σ − ζ 7) · x1, y7 = (σ − ζ )(σ − ζ 3)(σ − ζ 5) · x1

z1 = (σ − ζ 3)(σ − ζ 5)(σ − ζ 7) · x8, z3 = (σ − ζ )(σ − ζ 5)(σ − ζ 7) · x8

z5 = (σ − ζ )(σ − ζ 3)(σ − ζ 7) · x8, z7 = (σ − ζ )(σ − ζ 3)(σ − ζ 5) · x8.

It follows that

σ :yi 7→ ζ i yi , zi 7→ ζ i zi for i = 1, 3, 5, 7,

τ :y1 7→ −ζ 7z7, y3 7→ −ζ 5z5, y5 7→ −ζ 3z3, y7 7→ −ζ z1,

z1 7→ ζ 7 y7, z3 7→ ζ 5 y5, z5 7→ ζ 3 y3, z7 7→ ζ y1.

312 MING-CHANG KANG

If ρ(ζ ) = ζ a , then ρ(yi ) = yai and ρ(zi ) = zai for i = 1, 3, 5, 7. (The indexai is understood to be modulo 8.)

Apply Theorem 2.1. Then

K (ζ )(x1, . . . , x8)〈σ,τ,ρ〉

= K (ζ )(y1, y3, y5, y7, z1, z3, z5, z7)〈σ,τ,ρ〉

is rational provided that K (ζ ) (y1, y3, z5, z7)〈σ,τ,ρ〉 be rational when ρ(ζ )=ζ 3, that

K (ζ )(y1, y5, z3, z7)〈σ,τ,ρ〉 be rational when ρ(ζ )=ζ 5, and K (ζ )(y1, y7, z1, z7)

〈σ,τ,ρ〉

be rational when ρ(ζ )= ζ 7.

Subcase 6.1. ρ(ζ )= ζ 3.Define

u1 = y1z7, u2 = y3z5, u3 = y3/y31 , u4 = y8

1 .

Then K (ζ )(y1, y3, z5, z7)〈σ 〉

= K (ζ )(u1, . . . , u4) and the actions of τ and ρ aregiven by

τ : u1 7→ −u1, u2 7→ −u2, u3 7→ u2/u31u3, u4 7→ u8

1/u4,

ρ : u1 7→ u2, u2 7→ u1, u3 7→ 1/(u33u4), u4 7→ u8

3u34.

Define

r = u1, s = u2/u1, x = u1u3, y = u23u4/(u1u2), t = r(y − (1/y)).

Then

τ : r 7→ −r, s 7→ s, x 7→ −s/x, y 7→ 1/y, t 7→ t,

ρ : r 7→ rs, s 7→ 1/s, x 7→ 1/xy, y 7→ y, t 7→ st.

Note that K (ζ )(u1, . . . , u4)= K (ζ )(r, s, x, y)= K (ζ )(s, t, x, y).By Theorem 2.2, K (ζ )(s, t, x, y)〈τ 〉 = K (ζ )(s, t, u, v) where

u =

x +sx

xy +s

xy

, v =

y −1y

xy +s

xy

.

It is routine to check that ρ(u)= 1/u, ρ(v)= sv/u.Define

s ′= (ζ − ρ(ζ ))(1 + s)(1 − s)−1, t ′

= (1 + s)t,

u′= (ζ − ρ(ζ ))(1 + u)(1 − u)−1, v′

= (1 + (s/u))v.

We find that K (ζ )(s, t, u, v) = K (ζ )(s ′, t ′, u′, v′) and ρ(s ′) = s ′, ρ(t ′) = t ′,ρ(u′)= u′, ρ(v′)= v′. Thus K (ζ )(s, t, u, v)〈ρ〉

= K (s ′, t ′, u′, v′) is rational.

Subcase 6.2. ρ(ζ )= ζ 5.


Defineu1 = y1z7, u2 = y5z3, u3 = y5/y5

1 , u4 = y81 .

Then K (ζ )(y1, y5, z3, z7)〈σ 〉


τ : u1 7→ −u1, u2 7→ −u2, u3 7→ u2/u51u3, u4 7→ u8

1/u4,

ρ : u1 7→ u2, u2 7→ u1, u3 7→ 1/(u53u3

4), u4 7→ u83u5

4.

Define

r = u1, s = u2/u1, x = u22/(u

33u2

4), y = u1u23u4/u2, t = r(y − (1/y)).

Then

τ : r 7→ −r, s 7→ s, x 7→ s/x, y 7→ 1/y, t 7→ t,

ρ : r 7→ rs, s 7→ 1/s, x 7→ xy/s, y 7→ 1/y, t 7→ −st.

By Theorem 2.2, K (ζ )(u1, . . . , u4)〈τ 〉

= K (ζ )(s, t, x, y)〈τ 〉 = K (ζ )(s, t, u, v),where

u =

x −sx

xy −s

xy

, v =

y −1y

xy −s

xy

.

It is routine to check that ρ(u)= 1/u, ρ(v)= −sv/u.Define

s ′= (ζ − ρ(ζ ))(1 + s)(1 − s)−1, t ′

= (1 − s)t,

u′= (ζ − ρ(ζ ))(1 + u)(1 − u)−1, v′

= (1 − (s/u))v.

We find that K (ζ )(s, t, u, v)〈ρ〉= K (s ′, t ′, u′, v′) is rational.

Subcase 6.3. ρ(ζ )= ζ 7.Define

u1 = y1z7, u2 = y7z1, u3 = y7/y71 , u4 = y8

1 .

Then K (ζ )(y1, y7, z1, z7)〈σ 〉


τ : u1 7→ −u1, u2 7→ −u2, u3 7→ u2/u71u3, u4 7→ u8

1/u4,

ρ : u1 7→ u2, u2 7→ u1, u3 7→ 1/(u73u6

4), u4 7→ u83u7

4.

Define

r = u1, s = u2/u1, x = u3u4/u1, y = u22/(u

21u4

3u34), t = r(y − (1/y)).

314 MING-CHANG KANG

Then

τ : r 7→ −r, s 7→ s, x 7→ −s/x, y 7→ 1/y, t 7→ t,

ρ : r 7→ rs, s 7→ 1/s, x 7→ x/s, y 7→ 1/y, t 7→ −st.

By Theorem 2.2, K (ζ )(u1, . . . , u4)〈τ 〉

= K (ζ )(s, t, x, y)〈τ 〉 = K (ζ )(s, t, u, v),where

u =

x −Ax

xy −ABxy

, v =

y −By

xy −ABxy

with A = −s, B = 1.It is routine to check that

ρ : u 7→

x −Ax

Bxy

−Ayx

, v 7→

−s(

y −By

)Bxy

−Ayx

.

Define w = u/v. Then ρ(w)= −w/s. Note that

x −Ax

Bxy

−Ayx

=u

bu2 − Av2 ,

because this is the same identity as the identity (3–1) we encountered in Case 1.It follows that ρ(u)= u/(Bu2

− Av2)= C/u where C = w2/(w2+ s).

Definep = (1 − (1/s))w, q = (1 − s)t.

Then C = p2/(s + (1/s)+ p2− 2).

It follows that K (ζ )(s, t, u, v)〈ρ〉= K (ζ )(s, q, p, u)〈ρ〉, where

ρ : s 7→ 1/s, q 7→ q, p 7→ p, u 7→ C/u.

Define

X = s + (1/s), Y = (ζ − ρ(ζ ))(s − (1/s)),

Z = u + (C/u), W = (ζ − ρ(ζ ))(u − (C/u)).

Then K (ζ )(s, q, p, u)〈ρ〉= K (ζ )(s, p, u)〈ρ〉(q)= K (X, Y, Z ,W, p, q) because

[K (X, Y, Z ,W, p)(ζ ) : K (X, Y, Z ,W, p)] ≤ 2. The relations of X, Y, Z , W, pare given by

(3–2) 1X2− (Y/(ζ − ρ(ζ )))2 = 4


and

(3–3) Z2−(W/(ζ−ρ(ζ )))2 = 4C = 4p2/(s+(1/s)+p−2)= 4p2/(X+p2

−2).

Define η = 1/(ζ − ρ(ζ ))2 ∈ K . Then we find, from (3–2), that (X − 2)/Y =

ηY/(X + 2). From this we find that (X − 2)/Y , X and Y all lie in K ((X + 2)/Y ).Simplify (3–3). We get

(3–4) (Z(X − p2− 2)/(2p))2 − η(W (X − p2

− 2)/(2p))2 = X + p2− 2.

Let Z1 = Z(X − p2− 2)/(2p)− p, Z2 = Z(X − p2

− 2)/(2p)+ p, and W1 =

W (X − p2−2)/(2p). Thus (3–4) becomes Z1 Z2 −ηW 2

1 = X −2 ∈ K((X +2)/Y ).Thus Z2, p ∈ K((X +2)/Y, Z1,W1); hence K (X, Y, Z ,W, p, q)= K ((X +2)/Y,Z1,W1, q) is rational over K .

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Received to be supplied.

MING-CHANG KANG


NATIONAL TAIWAN UNIVERSITY

TAIPEI

TAIWAN

[email protected]


A MURASUGI DECOMPOSITION FOR ACHIRALALTERNATING LINKS

CAM VAN QUACH HONGLER AND CLAUDE WEBER

We prove that an achiral alternating link can be decomposed in a strongsense as a Murasugi sum of a link and its mirror image. The proof relieson our theory of Murasugi atoms. We introduce a notion of a bond betweenatoms, called adjacency. This relation is expressed by a graph, the adjacencygraph, which is an isotopy invariant. A well-defined link type, called a mol-ecule, is associated to any connected subgraph of the adjacency graph. TheFlyping Theorem of Menasco and Thistlethwaite is the main tool used toprove the isotopy invariance of atoms, molecules and the adjacency graph.The action of flypes on the adjacency graph and the invariance of the collec-tion of molecules under flypes are the main ingredients of the proof of thedecomposition theorem.

1. Introduction

A cheap way to produce an achiral knot is to construct the connected sum of a knotK with its mirror image; but it is certainly not true that all achiral knots arise inthis way! The main result of this article is that it is true that all alternating achiralknots (in fact, links) come from this procedure, if the notion of connected sum isreplaced by the more general notion of Murasugi sum.

As Mikami Hirasawa has kindly pointed out, we must clarify which kind ofMurasugi sum we work with. The classical definition of Murasugi sum involvestwo oriented links L ′ and L ′′ in S3. An arbitrary choice is made of Seifert surfacesS′ and S′′ for each link. Then on each surface a 2n-gon is chosen: P ′ in S′ andP ′′ in S′′. We select a 2-sphere S2 in S3 inducing a decomposition of S3 in twohemispheres. Then S′ is moved into one hemisphere and S′′ into the other in sucha way that, among other conditions,

P ′= S′

∩ S2= S′′

∩ S2= P ′′.

MSC2000: 57M25.Keywords: alternating links, achiral links, invertible links, Murasugi decomposition, flype

conjecture.We thank the Fonds National Suisse de la Recherche Scientifique for their support.

317

318 CAM VAN QUACH HONGLER AND CLAUDE WEBER

See [Gabai 1983] for more details. The abundance of possible choices has severalconsequences, most of them unwanted. For instance, Thompson [1994] has givenan example of two nontrivial knots which can be summed along adequately selectedSeifert surfaces (not of minimal genus) to produce the trivial knot. Hirasawa hasannounced, in a talk given in Geneva in February 2003, that given three orientedknots K0, K1 and K2 it is always possible to exhibit K0 as a Murasugi sum ofK1 and K2. On the other hand, Gabai [1983] has shown that if incompressible(or minimal genus) Seifert surfaces are used, then the resulting surface is again in-compressible (or of minimal genus). But even if one decides to restrict the choicesby imposing conditions on the Seifert surfaces, too many possibilities remain. Itseems hopeless to look for a uniqueness result for Murasugi decompositions in thiscontext. This is why, when we were working on our paper [Quach Hongler andWeber 2004], we decided to depart from Seifert surfaces and to deal with reducedalternating diagrams instead.

A diagrammatic Murasugi sum using these diagrams is defined in the beginningof Section 5, and we denote it by D′

∗ D′′. If we apply the Seifert construction, weobtain Seifert surfaces which have (among others) the property of being of minimalgenus. We take the liberty of writing L ′

∗ L ′′ for a sum of oriented links obtainedfrom reduced alternating diagrams. In this paper, we shall use only this refinedversion of Murasugi sums, and phenomena like those discovered by Thompsonand Hirasawa are thus impossible.

Notation. If L is an oriented link, L denotes the mirror image of L as an orientedlink. We write −L for the oriented link obtained from L by reversing the orienta-tions of all its components. An oriented link L is positively achiral if it is isotopicto L , and negatively achiral if it is isotopic to −L .

Theorem 5.2. (1) If L is positively achiral, then there exists a link L ′ such thatL = L ′

∗ L ′.

(2) If L is negatively achiral, then there exists a link L ′ such that L = L ′∗ −L ′.

The proof rests on our theory of Murasugi atoms [Quach Hongler and Weber2004]. In Section 2, we quickly recall basic facts about atoms, and then introducea notion of bond between atoms, which we call adjacency. Adjacency is a binaryrelation among atoms of a given link L which can be expressed by the adjacencygraph5(L). Its vertices are labeled by the various atoms of L and an edge connectstwo vertices if the corresponding atoms are adjacent. The graph is constructedfrom a reduced diagram D representing L , and the key point is to prove that theadjacency relation does not depend on the choice of D. The Flyping Theorem ofMenasco and Thistlethwaite [1993] is crucial here. As a consequence, the graph5(L) is an isotopy invariant. This is proved in Theorem 2.7 and Corollary 2.8.

A MURASUGI DECOMPOSITION FOR ACHIRAL ALTERNATING LINKS 319

Moreover, in Section 3, we associate to any connected subgraph of 5(L) a welldefined link type, which we call a molecule of L . We prove that the collection (aset with repetitions allowed) of molecules is an isotopy invariant. This collectionis a powerful tool for classification questions. In Section 4 we give applicationsof these concepts to reversibility and chirality questions. Theorem 5.2 is proved inSection 5. The proof uses the action of flypes on the adjacency graph studied inSection 2 and the invariance of molecules proved in Section 3.

Unless otherwise stated, by a link we mean an oriented, alternating, unsplittable,prime link in S3, and by a diagram we mean an oriented, alternating, connected,reduced, prime diagram in the 2-sphere S2 (not in the plane R2).

2. Adjacency

We recall briefly how atoms are obtained. For more details, see [Quach Honglerand Weber 2004], henceforth abbreviated [QW]. Let L be a link and let D be adiagram representing L . We perform a Seifert surgery at each crossing point of Dand obtain a bunch B of disjoint oriented circles in S2 called Seifert circles. Thesite of a surgery is indicated by an arc called the scar of the surgery. If each scaris endowed with a ± sign, it is easy to reconstruct D from B and the signed scars.This operation (inverse to surgery) is called suturing the scars. Each circle γ ∈ B

bounds two open discs 1′ and 1′′ in S2, called the Seifert discs determined by γ .The set of all Seifert discs determined by the various γ ∈ B is denoted by F. Thisset is ordered by inclusion, and a descending chain

10 ⊃11 ⊃ . . .⊃1k

is said to be of length k and to begin at 10.

Definition 2.1. A disc 1 ∈ F is of depth k if

(1) there exists a descending chain of length k beginning at 1, and

(2) there is no descending chain of length l beginning at 1 with l > k.

According to this definition, innermost discs correspond to discs of depth zero.Following Murasugi [1965], we call a Seifert bunch B special if each γ ∈B boundsan innermost disc.

We now define the Murasugi special components Di (for i = 1, 2, . . . , n) of D.First we look at the Seifert discs which are of depth one. Suturing the scars in eachsuch disc, we obtain disjoint, not necessarily prime, special diagrams D1, . . . , Dt .Then we remove from each disc of depth one the Seifert discs and scars it contains(we call this operation cleaning). We obtain a new diagram D′, a new bunch B′ anda new set of Seifert discs F′. From the Seifert discs of depth one in F′, we obtainnew special diagrams Dt+1, . . . , Ds . After cleaning, we obtain a new diagram


D′′, and so on. Finally the process comes to an end and we obtain the familyD1, . . . , Dn of Murasugi special components of D.

Although D is supposed to be prime, it may well be that Di is not. To makethings clear, recall that a Menasco circle for a diagram E is a Jordan curve 0 in S2

which cuts E transversally in two points, and such that each disc of S2 bounded by0 contains crossing points of E . By definition, a diagram is prime if there existsno Menasco circle for it. A beautiful theorem of Menasco [1984] says that if Eis prime then the link represented by E is also prime. The reason why there mayexist Menasco circles for Di is that Di has been obtained by cleaning many discs,thus allowing the Jordan curve 0 to have only two intersection points with Di , butmany more with D.

We define the prime factors Di1, . . . , Di

k of Di to be the subdiagrams of Di

which are maximal (with respect to inclusion) and prime. See Figure 1 for anexample.

We write D1, . . . , Ds for the diagrams of the prime factors of the various Mura-sugi components of D (the order is not important). We think of them as beingsubdiagrams of D and we call them the atom diagrams of D.

Definition 2.2. Let Da and Db be two atom diagrams of D. We say that they areadjacent if their union Da ∪ Db is again a diagram. As this union is necessarilyoriented and alternating, the condition is that it is connected and prime.

Lemma 2.3. The two atom diagrams Da and Db are adjacent if and only if :

(1) Da ∩ Db consists of exactly one Seifert circle γ ∈ B.

(2) Along γ the extremities of the scars of Da are interlaced with those of Db.

The proof is easy, once the following definition is stated.

Definition 2.4. Let X and Y be two disjoint, finite subsets of the circle S1. We saythat they are interlaced if there exist no disjoint intervals I ⊃ X and J ⊃ Y in S1.

Remark 2.5. Let 1′ and 1′′ be the two Seifert discs bounded by γ . Lemma 2.3and Definition 2.4 imply that Da ⊂1′ and Db ⊂1′′.

Definition 2.6. The adjacency graph 5(D) is the graph of the adjacency relationon the set of atom diagrams of D.

The graph 5(D) is simplicial. Its vertices are labeled by the atom diagrams ofD and two vertices are connected by an edge if the corresponding atom diagramsare adjacent. The process is illustrated in Figure 1; in this example the adjacencygraph 5(D) is

D1 D2 D3 D4.


D1

D2

D3

D4

Figure 1. Top left: the initial knot. Top right: the bunch withscars. Bottom: the corresponding atom diagrams.

Theorem 2.7. Let D and D∗ be two diagrams representing the same link L andsuppose that they differ by exactly one flype f . Then there exists a canonical iso-morphism ϕ f : 5(D) → 5(D∗) such that, for each atom diagram Da of D, theatom diagrams Da and ϕ(Da) represent the same link.

Proof. We shall go back to the proof of the topological invariance of atoms givenin [QW] and extract from it the fact that the adjacency relation is preserved undera flype. Recall that a flype starts from a tangle decomposition like this:

A B


Let α denote the Jordan curve which is the boundary of the tangle A.We ask the following question: How does α cut the Seifert bunch B associated

to the given diagram D?The answer is as follows. A Jordan curve α in S2 is the boundary of tangle A

giving rise to a tangle situation in exactly two cases.

Case 1. (i) α cuts one scar σ of B transversally,

(ii) α cuts transversally twice some Seifert circle γ of B and α does not cut B

somewhere else, and

(iii) α is nontrivial: each disc of S2 bounded by α contains some scars distinctfrom σ .

A priori there are two possibilities:

(1) No extremity of the the scar σ lies on γ .

(2) One extremity of the scar σ lies on γ .

Recall that the extremities of a scar always lie on different Seifert circles.Cases 1, 2, and 3 of [QW] correspond to the first possibility, while case 4 cor-

responds to the second one.

Case 2. The curve α contains a scar σ and cuts transversally two Seifert circles ofthe Seifert bunch B.

Case 5 of [QW] corresponds to this case.

The analysis carried out in [QW] reveals that, in all five cases examined there,the scar σ belongs to some special component, called the supporting componentof the flype. This component was denoted D0 but will be written E here, forconvenience.

We now proceed to analyse the prime components of E and their behaviourunder the flype. We shall do this by considering case 1 of [QW], but the argumentsare essentially the same in all five cases. We slightly modify Figure 10 of [QW]by sliding α along the scar σ to obtain the following:

A B0

I II

III IV V

VI

VII

VIII

Claim. If E is not prime, Menasco circles for E are entirely contained either in Aor in B.


Proof of claim. Think first of the preceding figure before any cleaning has beendone. By the definition of a flype, there are no scars outside tangles A and B exceptfor σ . Hence zones VI, VII and VIII are empty. Zones III, IV and V may eachcontain some scars and some Seifert circles. These have been removed during thecleaning process which leads to the picture of E . Therefore, each of zones III, IVand V may contain atom diagrams which are adjacent to a prime factor of E . Wenow look at zones I and II, as sketched in the figure. Both zones must contain discsand scars of E . This is because zone 0 and zone III must be connected throughzone I by scars and discs of E , or else the crossing of D which corresponds to thescar σ would be nugatory. For the same reason, zones IV and V must be connectedthrough zone II. �

We deduce from the above argument that the prime factors of E (namely theatom diagrams produced by E) are as follows:

(1) The atom diagram E0 which contains the scar σ , the two Seifert circles towhich σ is attached, the Seifert circle which is the boundary of (zone III ∪

zone IV ∪ zone VII), and some of the boundaries of colored discs and scarswhich are in zone I and in zone II.

(2) Maybe some other prime factors, each of them entirely contained in tangle Aor in tangle B. Denote these possible atom diagrams by Ei for i = 1, . . . , t .

We look again at the figure on the previous page and search for the atom diagramswhich are adjacent to E0 or to some Ei for i = 1, . . . , t . To do this, we considerthe initial bunch B and we perform cleaning operations in the coloured discs ofdepth 1 or greater, until they become exactly of depth 1. We thus obtain a bunchB′ and a corresponding diagram D′. We write G for an atom diagram of D′ whichis contained in a coloured disc.

Question. Which are the Gs adjacent to E0?

Answer. The Gs which are adjacent to E0 are:

(i) Those contained in zone 0, zone III, zone IV or zone V.

(ii) Some of the Gs contained in the coloured discs situated in A or B.

Question. Where are the G’s adjacent to some Ei for i = 1, . . . , t ?

Answer. (iii) They are in some coloured disc situated in A or in B.

What makes the difference between (ii) and (iii) depends on the position of theatom diagram G with respect to the Menasco circles for E . For each such Menascocircle µ j consider the disc M j bounded by µ j in tangle A or in tangle B. NowG is adjacent to E0 if it is outside all the M j and it is adjacent to some Ei if it isinside some M j .


We now consider the situation after the flype. The new diagram is D∗ withits corresponding bunch B∗. The supporting diagram is now the Murasugi spe-cial component of D∗ which contains the scar σ ∗. In [QW] we proved that thissupporting component is the image by the flype of the supporting component ofD. This is the equality (D∗)0 = (D0)

∗ of [QW]. We write here E∗ for the newsupporting component. The situation after the flype is this:

AB

0r

Ir II

IIIr IV V

VI

VII′ VII′′

VIII

We see that:

(1) Zone VII has disappeared, giving birth to zones VII′ and VII′′. This is of noconsequence because they are empty.

(2) The scar σ has, of course, disappeared. Zone 0 has undergone a 180-degreerotation and been connected to the big Seifert disc which contains the pointat infinity.

(3) Zones I and II have been turned.

We define (E∗)0 to be the prime factor of E∗ which contains the scar σ ∗.

Claim. The equality (E∗)0 = (E0)∗ holds, which allows us to write simply E0

∗.

Proof of claim. The equality can be seen by comparing the two preceding figures.Here are some details. In the figure above, E0

∗ is the atom diagram which contains

(a) the huge circle bounding the Seifert disc which contains the point at infinity,

(b) the boundary of (turned zone III) ∪ zone VII′,

(c) the boundary of zone IV ∪ zone VII′,

(d) the part of E0 which is in zone II, and

(e) the part of E0 which was in zone I, turned by a 180 degree rotation.

The key points from which the equality (E∗)0 = (E0)∗ can be deduced are:

(1) The Menasco circles which were in tangle B before the flype have not beenmoved.

(2) The Menasco circles which were in tangle A before the flype have simplybeen turned. �


We now construct the promised isomorphism

ϕ :5(D)→5(D∗)

We define the image of E0 by ϕ to be E0∗. As they differ by a flype, they represent

the same atom.Any other atom diagram of D is entirely contained in either tangle A or in tangle

B. If it is contained in B, then ϕ sends it to itself. If it is in A, then ϕ sends it to itsimage under the rotation. From this, we immediately deduce that if two adjacentatom diagrams of D are entirely contained in A or in B, the adjacency relation ispreserved by ϕ.

It remains to see what happens to atom diagrams of D which are adjacent toE0. For those which are contained in B (that is, in zones II, IV and V) their imageunder ϕ will be adjacent to E0

∗ because nothing changes in B under the flype.Those which are contained in A are turned by the flype. For those in zones 0 andIII, their image will clearly be adjacent to E0

∗. This is also true for those in zoneI, because their image under the rotation lies outside the image of all the Menascocircles. This completes the proof of Theorem 2.7. �

Corollary 2.8. Let D and D′ be two diagrams representing the same link type L .Then 5(D) and 5(D′) are isomorphic.

Proof. The main result of [Menasco and Thistlethwaite 1993] asserts that D andD′ differ by a finite sequence of flypes and homeomorphisms of S2. �

Remark. Corollary 2.8 says that there exists an isomorphism 9 between 5(D)and 5(D′) such that, for each vertex Da of 5(D), the atom diagrams Da and9(Da) represent isotopic links. It would be pleasing to have a canonical isomor-phism between the two adjacency graphs, but in general this is not possible becausediagrams can have nontrivial automorphisms. To amend this state of affairs, wecan categorize. Here are some brief details.

Let 0 and 0′ be two graphs embedded in S2. A map 8 : (S2, 0)→ (S2, 0′) is ahomeomorphism 8 : S2

→ S2 of degree +1 such that 8(0)= 0′. An isotopy is acontinuous family 8t of maps (S2, 0)→ (S2, 0′) for t ∈ [0, 1]. An isotopy classof maps will be called a spherical equivalence.

The difficulty about the non-canonicity of9 is that the group of spherical equiv-alences of a diagram in S2 can be nontrivial. The following easily proved lemmawill help.

Lemma 2.9. Let 0 be a connected graph in S2 and let e be an oriented edge of 0.Then a spherical equivalence 8 : (S2, 0) → (S2, 0) such that 8(e) = e keepingthe orientation of e fixed, is the identity.

As an application of the lemma, consider a diagram D and its image D∗ by aflype. Now, D∗ is actually not a well defined graph embedded as a subspace of S2,


as several choices are involved. But any two representatives for D∗ are canonicallyisomorphic, because a flype is the identity on some part of D.

With this in hand, we can consider the category whose objects are diagrams inS2 and whose morphisms are flypes. Then the adjacency graph is a functor5 fromthis category to the category of simplicial graphs, with labeled vertices. The labelsare the atom diagrams. This is a sophisticated way to state that the adjacency graphis an isotopy invariant.

However, we shall often dare to speak of the adjacency graph 5(L) of a link L .The vertices of 5(L) are then labeled by the atoms of L .

Proposition 2.10. The adjacency graph 5(L) of a link L is bipartite: adjacentvertices represent prime special links of opposite sign.

Proof. Let D be a special diagram. Then all crossings of D have the same Conwaysign. We have seen above (see Remark 2.5) that if Da and Db are two adjacentatom diagrams, there exists a Seifert circle γ such that Da ∩ Db = γ . If 1′ and 1′′

are the two Seifert discs bounded by γ , then Da ⊂1′ and Db ⊂1′′. As Da ∪ Db

is alternating, this implies that the sign of Da is the opposite of the sign of Db.Now, the sign of a special diagram is preserved by a flype. Hence the iso-

morphism ϕ f : 5(D) → 5(D′), associated to a flype f , preserves the signs ofthe vertices. As a consequence, atoms have signs and two atoms which are theextremities of an edge have opposite signs. �

We wish to produce some examples. This raises a problem of notation, becausethe knots pictured in the tables are not oriented. This is not too consequential forthe examples given below, because the knots we shall use have few crossings andhence are usually reversible. The situation is worse for links. We shall deal moreextensively with these matters in Section 4. We write the sign as an exponent,hence 31

+ denotes a trefoil with positive Conway signs.

Example 2.11. The knots 1042 and 1043 have the same collection{22

1+, 22

1−, 3+

1 , 3−

1

}of atoms. Their unlabeled adjacency graph is

A B C D

in both cases. The vertices are labeled

A = 3+

1 , B = 3−

1 , C = 221+, D = 22

1−

for the knot 1042, and

A = 3+

1 , B = 221−, C = 22

1+, D = 3−

1

for the knot 1043.


We see that the graphs with vertices labeled by the atoms are non-isomorphic.Incidentally, this shows that the two knots are distinct.

Not surprisingly, the adjacency graph is too weak to distinguish knots. Forinstance, the knots 10115 and 1043 have the same adjacency graph. The smallestsuch examples are provided by the graph 3+

1 3−

1 which corresponds to the knot63 but also to the link 63

2 and by the graph 221+ 42

1− which corresponds to the

knot 62 and to the link 631 for some orientation.

3. Molecules

Proposition 3.1. Let L be a link and let D and D′ be two diagrams for L differingby a flype f . Let H ⊂ 5(D) be a connected subgraph of 5(D) and let L(H) bethe link represented by the union of the atom diagrams which correspond to thevertices of H . Let ϕ f :5(D)→5(D′) be the canonical isomorphism induced bythe flype f . Then L(H) and L(ϕ(H)) represent the same link type.

As a consequence, again thanks to the Menasco–Thistlethwaite Flyping Theo-rem, if L is a given link, we can associate to any connected subgraph of 5(L) awell defined link type. The collection of links obtained this way from the set ofconnected subgraphs of 5(L) will be called the collection of molecules of L andwe shall denote it by M(L). As for the collection of atoms, we remark that a givenlink type may appear several times in the list. Notice that molecules are primelinks, as they correspond to connected adjacency subgraphs.

Warning. If we want to point out to a particular molecule, “the” graph 5(L)may be too imprecise. This is especially the case when “the” adjacency graph hasnontrivial automorphisms (as a labeled graph). Then it is better to use the moreelaborate categorical setting sketched above.

Example 3.2. A necessary condition for two links L and L ′ to be isotopic is thatM(L)= M(L ′). Consider the knots 1045 and 1088. Both have the same adjacencygraph (and hence the same atoms)

221+

3−

1 3+

1 221−.

However the collection of molecules distinguishes the two knots, because the mol-ecule associated to the subgraph 3−

1 3+

1 is the knot 63 for 1045 and “the” link63

2 for 1088.

Proof of Proposition 3.1. The detailed proof of Theorem 2.7 contains all the ingre-dients necessary to prove Proposition 3.1. Two possibilities can occur:

If H contains a prime factor of the supporting diagram of the flype f , thenL(ϕ(H)) is represented by H∗, the image of H under the flype.


If H does not contain a prime factor of the supporting diagram, then its imageis translated and/or turned. �

We now consider the molecules which are built up from two adjacent atomdiagrams. In fact, the existence of an edge between two vertices says that somebond exists between the two corresponding atoms. The edge in itself says nothingabout the nature of the bond, but the two-atom molecule reveals what this bond is.We call such molecules edge-molecules.

One can then add a weight on the edges of the adjacency graph, the weight beingthe edge-molecules; we call the resulting graph the bond graph. It is very temptingto conjecture that a link is determined by its bond-graph, but this is (alas?) wrong.An example is provided by the graph

221+

3−

1 221+.

The edge-molecules are the knot 52. But the knot 77 and the link 731 (for some

orientation) both have this weighted graph as bond graph.Another temptation is to look at the vertex stars. By definition, if P is a vertex

in a simplicial graph 5, the star st(P) is the subgraph of 5 which is the union ofthe edges (and their extremities) which have P as a vertex. Accordingly, if L i isthe atom which corresponds to a vertex of5(L) the molecule associated to the starst(L i ) describe a kind of neighborhood of L i in L .

Question. Let L and L ′ be two links and suppose that there exists an isomorphismϕ : 5(L) → 5(L ′) such that, for any atom L i of L , st(L i ) and st(ϕ(L i )) areisotopic. Are L and L i isotopic?

The answer is, in general, no (mutations can be used to construct counter-examples) but the answer might be yes if the graph is 2-connected.

4. Applications to chirality and reversibility

If L is a link, we denote by L the mirror image of L as an oriented link. We denoteby −L the oriented link obtained by changing the orientation of all the componentsof L . The same notations will be used for diagrams.

If5(L) is the adjacency graph of L , the graph obtained from5(L) by replacingthe label L i at each vertex by the label L i will be written 5(L). Clearly 5(L) =

5(L). Analogously −5(L) denotes the adjacency graph obtained from 5(L) byreplacing each vertex label L i by −L i . Obviously one has −5(L)=5(−L). Werecall now the usual definitions.

Definition 4.1. A link L is positively achiral if it is isotopic to L , negatively achiralif it is isotopic to −L , and reversible if it is isotopic to −L .


Theorem 4.2. If the link L is positively achiral, the graph 5(L) is isomorphic to5(L). If it is negatively achiral, the graph 5(L) is isomorphic to −5(L). If it isreversible, the graph 5(L) is isomorphic to −5(L).

Proof. The proof is an immediate consequence of the isotopy invariance of theadjacency graph, loosely stated. �

We write M(L) for the collection of molecules of the link L . Analogously, wewrite −M(L) and −M(L) for the molecule collection of, respectively, −L and −L .

Corollary 4.3. (1) If L is positively achiral, then M(L)= M(L).

(2) If L is negatively achiral, then M(L)= −M(L).

(3) If L is reversible, one has M(L)= −M(L).

We now produce some examples. As far as chirality questions are concerned, itis easy to produce many, because atoms are chiral. For instance, consider the knot1042 whose adjacency graph

3+

1 3−

1 221+

221−

was introduced in Example 2.11. We see that the atoms can be grouped in pairs(D, D) so that the knot passes the first test for achirality. However, the graph5(D)is not isomorphic to 5(D) and hence 1042 is chiral.

Reversibility questions are more delicate to handle, because atoms can be re-versible or non-reversible. In each case, an ad hoc proof is needed. Here is anexample of a diagram D for a 3-component link:

The adjacency graph 5(D) is

221+

421−

421+

221−.

The edge molecule associated to the middle bond is the knot 817 which is knownto be non-reversible. This implies that the link is also non-reversible.


5. Decomposition of achiral links as a Murasugi sum

Definition 5.1. Let L be a link and {M1, . . . ,Ms} be a collection of moleculeswhose atoms constitute a partition of the collection of atoms of L . Let D be a dia-gram for L and let D1, . . . , Ds be the subdiagrams of D representing the moleculesM1, . . . ,Ms . Suppose that all the Di have the same Seifert circle γ in common.Then we say that the molecule diagrams D1, . . . , Ds form a decomposition of Das a diagrammatic Murasugi sum.

The pioneering work of Murasugi shows that the link L is a Murasugi sum, inthe classical sense, of the links M1, . . . ,Ms . The Seifert circle γ will be calleda plumbing (Seifert) circle. Traditionally one considers Murasugi sums with twofactors, and this can easily be achieved at the cost of dealing with possibly non-prime link factors. For instance, one can choose an integer k such that 1 < k < sand consider the subdiagrams D′

=⋃

i=1,...,k Di and D′′=⋃

i=(k+1),...,s Di . If L ′

is the link represented by D′ and if L ′′ is the link represented by D′′, then L is aMurasugi sum L = L ′

∗ L ′′ with two factors.The aim of this section is to prove the following theorem.

Theorem 5.2. (1) If L is positively achiral, there exists a link L ′ such that L =

L ′∗ L ′.

(2) If L is negatively achiral, there exists a link L ′ such that L = L ′∗ −L ′.

Remark. The decomposition is not necessarily unique and the link L ′ is not nec-essarily prime. How this can happen is explained below; see the proof of Theorem5.2 when 5(D) is 2-connected.

Proof. We prove the theorem when L is positively achiral. If L is negativelyachiral, it suffices to replace 5(D) by −5(D) in the proof given below.

Let D be a diagram for L . We consider the graphs 5(D) and 5(D). By[Menasco and Thistlethwaite 1993] and Theorem 2.7 there exists a sequence offlypes which induces an isomorphism from 5(D) to 5(D). As such a sequence isnot necessarily unique, we choose one and write

8 : 5(D)→5(D)

for the induced isomorphism.We now write 5(D) for the graphs5(D) or 5(D) with no label at the vertices.

Then 8 induces an isomorphism of (plain) graphs

8 : 5(D)→ 5(D).

Remark 5.3. No vertex of 5(D) is fixed by 8, as 8 sends a vertex of 5(D) toa vertex of 5(D) with the same label. As a consequence, no edge of 5(D) ispointwise fixed by 8.


Proof of Theorem 5.2 when 5(D) is a tree. By the Hopf–Lefschetz fixed pointtheorem, 8 has fixed points. Remark 5.3 implies that 8 has a unique fixed point,and that this fixed point is located in the middle of an edge e. Now remove theinterior of e. Then 5(D) is split into two subtrees 5(D)′ and 5(D)′′. Similarly,5(D) is split into 5(D)′ and 5(D)′′. As 8 exchanges the extremities of the edgee, 8 induces an isomorphism of 5(D)′ onto 5(D)′′, and of 5(D)′′ onto 5(D)′.The topological invariance of molecules proved in Proposition 3.1 implies that themolecule L ′ associated to5(D)′ is the mirror image L ′ of the molecule associatedto 5(D)′′.

On the other hand, the edge e has two vertices V ′∈5(D)′ and V ′′

∈5(D)′′. Theedge between V ′ and V ′′ indicates a Murasugi sum between the atom representedby V ′ and the atom represented by V ′′. Indeed, the adjacency relation representedby e says precisely that the two atoms have a common Seifert circle γ . But thisargument also shows that we have a diagrammatic Murasugi sum between 5(D)′

and 5(D)′′. As a consequence, L is a Murasugi sum of the molecule L ′ and themolecule L ′. �

In order to be able to argue when 5(D) is not a tree, we need some facts aboutsimplicial graph theory. A general reference for the results we need is provided bythe book [Diestel 2000]. A circuit is a graph which is homeomorphic to a circle. Agraph G is 2-connected if it contains at least three vertices and if any two distinctvertices are situated on at least one circuit. By Menger’s theorem, a connected G is2-connected if and only if it contains no cut vertex and is not the connected graphwith only one vertex or only one edge.

Proposition 5.4. Suppose that 5(D) is 2-connected. Then all atom diagrams of Dhave one Seifert circle γ in common.

Proof of Proposition 5.4. For the moment, let D be any diagram (not necessarilyprime) and let γ be a Seifert circle of D. Let 1′ and 1′′ be the two open discs ofS2 bounded by γ , and let 1′ and 1′′ be their closure. We define the depth dγ ofγ to be the pair of integers

(k ′γ , k ′′

γ

)where k ′

γ is the depth of 1′ and k ′′γ the one of

1′′. After a possible change of notation we can assume that 0 ≤ k ′γ ≤ k ′′

γ .

Remark. If k ′γ = 0 for all γ in the bunch B, then k ′′

γ = 1 for all γ ∈ B and, in fact,the diagram D is special. In this case, the graph 5(D) has no edge. It is a disjointunion of vertices, one vertex for each atom diagram. However, if there are morethan one atom present, one can consider that D is a diagrammatic Murasugi sum,as it is a connected sum.

As a consequence, if 5(D) is 2-connected, there must exist Seifert circles γwith k ′

γ ≥ 1.

Claim. If 5(D) is 2-connected, and if γ ∈ B such that k ′γ ≥ 1 then k ′

γ = k ′′γ = 1.


Proof of claim. Suppose that k ′′γ ≥ 2. Choose a descending sequence

1′′

γ ⊃11 ⊃ · · · ⊃1n

of Seifert discs, with n ≥ 2. Let γi be the boundary of 1i . Consider the uniqueatom diagram D0 which has γ and γ1 as Seifert circles. (It is easy to see thatgiven any two Seifert circles of a Seifert bunch, there is at most one atom diagramwhich contains these two circles.) As k ′

γ ≥ 1, there are atom diagrams in 1′, andby construction there are atom diagrams in 11. As D0 is unique, if one removesthe vertex D0 from the graph 5(D), there is no way to connect, in the remaininggraph, atom diagrams in 1′ to atom diagrams in 11. In other words, D0 is a cut-vertex of 5(D), which contradicts the assumption that 5(D) is 2-connected. �

Now we observe that, in any connected diagram, there cannot exist more thanone Seifert circle γ with dγ = (1, 1). Let γ be the unique such Seifert circle forD. All atom diagrams contain γ . This completes the proof of Proposition 5.4. �

Proof of Theorem 5.2 when 5(D) is 2-connected. We know from Proposition 5.4that there exists exactly one Seifert circle γ with dγ = (1, 1). We claim that γ isthe plumbing circle we seek. Let 1′ and 1′′ be the two open discs bounded by γin S2. Let D1

′, . . . , Ds′ be the atom diagrams which are in1′ and have γ as one of

their Seifert circles, and let D1′′, . . . , Dt

′′ be those which are in 1′′. In fact, thereare no more atoms in D, otherwise γ would not be of depth (1, 1). The atoms in1′

have all the same sign (say +) and those in 1′′ have all the opposite sign (say −).

Note. This argument shows that every 2-connected adjacency graph is a subgraphof the complete bipartite graph on (s, t) vertices. We do not know which subgraphscan actually be obtained this way. It is easy to realize the complete graphs.

Now, the isomorphism8 exchanges the atoms which are in1′ with those whichare in 1′′, and hence s = t . Let L ′

i be the link represented by D′

i and, similarly,let L ′′

j be the link represented by D′′

j , for i = 1, . . . , s and j = 1, . . . , s. After apossible change of numbering we can assume that 8

(D′

i

)= D′′

i for i = 1, . . . , s.As 8 is induced by flypes, L ′

i is isotopic to L ′′

i for i = 1, . . . , s. Write L ′ for theconnected sum of the L ′

i for i = 1, . . . , s, and L ′′ for the connected sum of the L ′′

ifor i = 1, . . . s. Then L ′ is isotopic to L ′′ and L is a Murasugi sum of L ′ and L ′′. �

We now need a slight modification of the block decomposition of a graph. Toavoid confusion we shall use the word “brick”.

Definition 5.5. A brick of a connected graph G is a subgraph of G which is 2-connected and maximal (with respect to inclusion) for this property.

Let {B1, . . . , Bs} be the bricks of the connected graph G. If a 6= b then Ba ∩ Bb

is either empty or consists of just one vertex. Let


A = G \

s⋃i=1

Int(Bi ).

A is a forest, that is, a disjoint union of trees. Let

A =

t∐j=1

A j ,

where each A j is a tree. We now perform the following enlargement on G.Let Ba be a brick. Then the intersection Ba ∩ A is a finite set of vertices

{P1, . . . , Pu}. Consider the disjoint union

(G \ Int Ba) q Ba.

Each vertex P1, . . . ,Pu appears once in (G \ Int Ba) and once in Ba . From thedisjoint union we construct a new graph by joining both appearances of Pi by anew edge for i = 1, . . . , u. We perform the same operation for each brick Ba . Atthe end, we get a connected graph G. Here is an example:

G

→

G

G contains canonically each brick Bi and each tree A j . In G one has Bi ∩Bk =∅if i 6= k and A j ∩ Bi = ∅ for all i = 1, . . . , s and all j = 1, . . . , t . In G we nowcontract separately to a point each brick Bi . This quotient graph is written G:

G


It is easy to prove that G is a tree. This tree has two kinds of vertices. Firstly,the vertices of the forest A, which we call ancient vertices. Secondly, the verticeswhich correspond to the smashed bricks, denoted by a star on the figure above, andwhich we call brick vertices.

Suppose now that 9 : G → G is a graph automorphism. The construction of Gand of G is so natural that 9 induces automorphisms 9 : G → G and 9 : G → G.The automorphism 9 sends a brick vertex to a brick vertex and an ancient vertexto an ancient one.

Proof of Theorem 5.2 in the general case. We apply the bar construction above tothe graph G = 5(D). We write 5(D) for the graph thus obtained from 5(D). As5(D) is a tree, the automorphism 8 has fixed points.

Claim. 8 has exactly one fixed point. It is either the mid-point of an edge whoseextremities are two ancient vertices or a brick vertex.

Proof of claim. By construction, the vertices of an edge of 5(D) are either twoancient vertices or one brick vertex and one ancient vertex. As every ancient vertexis moved by 8, no edge is pointwise fixed by 8. This implies that 8 has exactlyone fixed point and that this fixed point is of one of the two forms as stated. �

Suppose that the fixed point is the mid-point of an edge e. Let e be the edge of5(D) which projects onto e. This edge e is a bridge of 5(D). The automorphism8 : 5(D) → 5(D) has the mid-point of e as fixed point. From here, the proofproceeds in the same way as when 5(D) is a tree. We remove the interior of efrom 5(D) which is thus split into two connected subgraphs permuted by 8 andso on.

Suppose that the fixed point is a brick vertex. By construction, this means thatthere is a brick Ba which is invariant by 8. Remove the fixed point from 5(D).The remaining graph is a disjoint union of trees and these trees are permuted by8. No tree is invariant, as there is only one fixed point. Hence, we can find apartition

{A1,8(A1), . . . , An,8(An)

}of the set of trees. Now consider the graph

5(D) and remove from it the interior of the brick Ba . The remaining graph is adisjoint union of subgraphs. We write G j for the subgraph which projects ontoA j (for j = 1, . . . , n). The remaining subgraph is indeed the disjoint union of{G1, 8(G1), . . . ,Gn, 8(Gn)

}. The intersection G j ∩ Ba is a cut vertex Pj of

5(D).We return to the proof of Theorem 5.2 when 5(D) is 2-connected. Consider

the brick Ba of 5(D). Let γ be the plumbing circle for Ba . Let 1′ and 1′′ bethe Seifert discs bounded by γ as above. After a possible change of notation, wecan assume that the diagram D j which corresponds to G j is contained in 1′ forj = 1, . . . , n. Hence, the diagram which corresponds to 8(G j ) is contained in1′′.


The diagram which corresponds to the cut vertex Pj is one of the atom diagrams ofBa which are in 1′. The rest of the proof now follows as in the case where 5(D)is 2-connected. �

Example. Consider the following 3-component link L with its diagram D:

42+

1 22−

1

22+

142−

1

Its graph 5(D) is 2-connected. The link L can be considered as a Murasugi sumof a 6-crossing link which is the connected sum of a 22

1+ and a 42

1+ with its mirror

image. It can be also considered as a Murasugi sum of the prime knot 6+

2 with itsmirror image.

Remark 5.6. The simplicity of the proof of Theorem 5.2 when5(D) is a tree madeone of us (Quach) believe very early that the theorem could be true in general. Butif circuits are present in 5(D) it is possible that 8 has no fixed points. It is herethat graph theory comes to the rescue. Circuits are organized in bricks and thegraph of bricks is a tree. As the theorem is true when 5(D) is 2-connected, thegeneral case follows. A fixed point is absolutely needed in our proof, because itindicates where the plumbing Seifert circle is to be found.

Acknowledgements

We wish to thank Morwen Thistlethwaite for several interesting conversations, andthe Fonds National Suisse de la Recherche Scientifique for their support.

References

[Diestel 2000] R. Diestel, Graphentheorie, Springer-Lehrbuch, Springer, Berlin, 2000. MR 1411445Zbl 0957.05001

[Gabai 1983] D. Gabai, “The Murasugi sum is a natural geometric operation”, pp. 131–143 inLow-dimensional topology (San Francisco, Calif., 1981), Contemp. Math. 20, Amer. Math. Soc.,Providence, RI, 1983. MR 85d:57003 Zbl 0524.57004

[Menasco 1984] W. Menasco, “Closed incompressible surfaces in alternating knot and link comple-ments”, Topology 23:1 (1984), 37–44. MR 86b:57004 Zbl 0525.57003


[Menasco and Thistlethwaite 1993] W. Menasco and M. Thistlethwaite, “The classification of alter-nating links”, Ann. of Math. (2) 138:1 (1993), 113–171. MR 95g:57015 Zbl 0809.57002

[Murasugi 1965] K. Murasugi, “On a certain numerical invariant of link types”, Trans. Amer. Math.Soc. 117 (1965), 387–422. MR 30 #1506 Zbl 0137.17903

[Quach Hongler and Weber 2004] C. V. Quach Hongler and C. Weber, “On the topological invari-ance of Murasugi special components of an alternating link”, Math. Proc. Cambridge Philos. Soc.137:1 (2004), 95–108. MR 2005h:57011 Zbl 1057.57007

[Thompson 1994] A. Thompson, “A note on Murasugi sums”, Pacific J. Math. 163:2 (1994), 393–395. MR 94k:57018 Zbl 0809.57003

Received January 10, 2004. Revised January 13, 2005.

CAM VAN QUACH HONGLER

SECTION DE MATHEMATIQUES

UNIVERSITE DE GENEVE

CP 64CH-1211 GENEVE 4SWITZERLAND

[email protected]

CLAUDE WEBER

SECTION DE MATHEMATIQUES

UNIVERSITE DE GENEVE

CP 64CH-1211 GENEVE 4SWITZERLAND

[email protected]


A NEW BOUND FOR FINITE FIELD BESICOVITCH SETSIN FOUR DIMENSIONS

TERENCE TAO

Let F be a finite field with characteristic greater than two. A Besicovitchset in F4 is a set P ⊆ F4 containing a line in every direction. The Kakeyaconjecture asserts that P and F4 have roughly the same size, in the sensethat |P|/|F|4 exceeds Cε|F|−ε for ε > 0 arbitrarily small, where Cε doesnot depend on P or F. Wolff showed that |P| exceeds a universal constanttimes |F|3. Here we improve his exponent to 3+

116 −ε for ε > 0 arbitrarily

small. On the other hand, we show that Wolff’s bound of |F|3 is sharp if werelax the assumption that the lines point in different directions. One newfeature in the argument is the use of some basic algebraic geometry.

1. Introduction

Let F be a finite field with characteristic greater than 2. For any n ≥ 2, we define aBesicovitch set in Fn to be a set P ⊆ Fn containing a line in every direction (everyequivalence class under parallelism). The finite field Kakeya conjecture (see [Wolff1998b], for example) asserts that |P| ≥ Cε|F |

n−ε for any ε > 0, where |P| denotesthe cardinality of P and the quantities Cε are independent of |F |. This conjectureis the finite field analogue of the Euclidean Kakeya set conjecture, which is relatedto several other problems in harmonic analysis; see [Wolff 1998b; Mockenhauptand Tao 2004] for further discussion on this. Basically, one can view the finite fieldKakeya problem as a simplified model for the more interesting Euclidean Kakeyaproblem, where technical difficulties involving small separations, small angles, andmultiple scales complicate the task (as discussed briefly in Section 9).

Informally, the Kakeya conjecture asserts that lines pointing in different direc-tions in Fn cannot have substantial overlap. This conjecture has been proved intwo dimensions but remains open in higher dimensions. In [Wolff 1998b] (see also[Wolff 1995; Mockenhaupt and Tao 2004]) it was shown that |P| & |F |

(n+2)/2,

MSC2000: 42B25, 05C35.Keywords: Besicovitch sets, affine spaces over finite fields, Kakeya conjecture, reguli.This work was inspired by the April 2002 Instructional Conference on Combinatorial aspects ofMathematical Analysis at University of Edinburgh. The author is a Clay Prize Fellow and is sup-ported by the Packard Foundation.

337

338 TERENCE TAO

where A & B means that A ≥ C−1 B for some universal constant C . In fact, morewas proved:

Definition 1.1. A family L of lines in Fn is said to obey the Wolff axiom if forevery 2 ≤ k ≤ n − 1, every k-dimensional affine subspace V ⊂ Fn contains atmost O(|F |

k−1) lines in L . (Here we view the field F as being quite large, andthe family L as depending on F . The implied constant in the O( ) notation maydepend on n and k but is uniform in F . Also Recall that an affine subspace is atranslate of a vector subspace of Fn .)

Theorem 1.2 [Wolff 1995; 1998b]. If L is a family of O(|F |n−1) lines obeying the

Wolff axiom, and P ⊆ Fn contains all the lines in L, then |P| & |F |(n+2)/2.

In fact one only needs to use the Wolff axiom for k = 2. From this theoremand the observation that any family of lines that point in different directions auto-matically obeys the Wolff axiom, we immediately see that Besicovitch sets havecardinality & |F |

(n+2)/2.In [Mockenhaupt and Tao 2004] (see also [Katz et al. 2000]) it was observed

that the statement of Theorem 1.2 is sharp in three dimensions, in the sense thatthere exist finite fields F and collections of lines L in F3 obeying the Wolff axiomand a collection P of points containing all the lines in L , such that |P| ∼ |F |

5/2

(where A ∼ B means that A & B and B & A). Indeed, if F contains a subfield Gof index 2, with the accompanying involution z 7→ z on F , one can take P to bethe Heisenberg group

P := {(z1, z2, z3) ∈ F3: Im(z3)= Im(z1 z2)},

where Im(z) := (z− z)/2. (It is an interesting question whether an example similarto this can be obtained if F does not contain a subfield of index 2.)

Our first observation is that Theorem 1.2 is also sharp in four dimensions:

Proposition 1.3. Let 〈 , 〉 : F4× F4

→ F be a nondegenerate symmetric quadraticform on F4. Let P be the “unit sphere”

(1–1) P := {x ∈ F4: 〈x, x〉 = 1}

and let L be the set of all lines of the form {x+tv : t ∈ F}, where x ∈ F4, v∈ F4\{0}

are such that 〈x, x〉 = 1, 〈v, x〉 = 0, and 〈v, v〉 = 0. Then L has cardinality|L| ∼ |F |

3 and obeys the Wolff axiom, while P has cardinality |P| ∼ |F |3 and

contains all the lines in L.

We prove this in Section 3. A similar counterexample can be created in R4

as long as one chooses the form 〈 , 〉 to be indefinite. The proposition does notcontradict the Kakeya conjecture because the lines L do not all point in differentdirections (despite obeying the Wolff axiom). Nevertheless, it seems of interest

A NEW BOUND FOR BESICOVITCH SETS IN FOUR DIMENSIONS 339

to extend this example (and the Heisenberg group) to higher dimensions, thoughperhaps the bound of |F |

(n+2)/2 in Theorem 1.2 need not be sharp for large n.This example illustrates two things. Firstly, in order to progress toward the

Kakeya conjecture in low dimensions one must make better use of the hypothesisthat the lines in L point in different directions; merely assuming the Wolff axiomwill not by itself suffice. (In high dimensions — say n ≥ 9 — there are other, more“arithmetic” arguments available to improve upon Theorem 1.2. See [Bourgain1999; Katz and Tao 1999; 2002b; Rogers 2001; Mockenhaupt and Tao 2004].)

Secondly, the algebraic geometry of quadric surfaces may be relevant to theKakeya problem.1 In the three-dimensional case n = 3, quadric surfaces are essen-tially the same thing as reguli, those ruled surfaces consisting of all the lines thatintersect three fixed lines in general position. In particular, we have the “three-linelemma”, which asserts that given three mutually skew lines in F3, there are at mostO(|F |) lines in different directions that intersect all three.

Reguli have already come up in the work of Schlag [1998], who used the three-line lemma to give a new proof of Bourgain’s estimate [1991]

(1–2) |P| & |F |7/3

in three dimensions. While it is true that this bound has since been superseded bythe estimate in Theorem 1.2, we shall need to follow [Schlag 1998] and make useof reguli and the three-line lemma in what follows. We are indebted to Nets Katzfor pointing out the usefulness of reguli in the low-dimensional Kakeya problem.Indeed, our work here was inspired by similar work in three dimensions by NetsKatz (currently in preparation).

The main result of this paper is the following improved bound on the cardinalityof Besicovitch sets in four dimensions. We use A / B to denote the estimateA ≤ Cε|F |

εB for any ε > 0, where Cε is a quantity depending only on ε.

Theorem 1.4. If P is a Besicovitch set in |F |4, then |P| ' |F |

3+1

16 .

One can probably improve the ' to a & by going through the argument in thispaper more carefully, but we will not do so here in order to simplify the exposition.

The paper is organized as follows. After setting out our incidence geometrynotation in Section 2, we prove Proposition 1.3 in Section 3. We then review somebasic algebraic geometry in Section 4, culminating in a “three-regulus lemma”in F4, which will be the analogue of the three-line lemma in F3. In Section 5we review some combinatorial preliminaries, before starting the proof of Theorem1.4, which occupies the next three sections. The first step is to use a standard

1There seems to be a parallel phenomenon in recent work on Szemerédi’s theorem on arithmeticprogressions, in that while arithmetic progressions are rather “linear” quantities, they give rise rathernaturally to other “quadratic” objects which then need to be studied. See [Gowers 1998].

340 TERENCE TAO

“iterated popularity” argument (as in [Christ 1998], for example), together witha rudimentary version of the “plate number” argument in [Wolff 1998a], in orderto refine the Besicovitch set to a uniform, nondegenerate collection of points andlines. After a sufficient number of refinements, we can construct a large number ofreguli incident to many lines in the Besicovitch set, and eventually get about |F |

3

lines incident to three distinct reguli (if |P| is too close to |F |3); this will contradict

the three-regulus lemma mentioned earlier.

2. Incidence notation

We now set some notation for the finite field geometry of the affine space F4. Aline in F4 is a set of the form l = {x + tv : t ∈ F} where x, v ∈ F4 and v is nonzero.Two lines are parallel if they are translates of each other but not identical; a set oflines is said to point in different directions if no two lines in the set are parallel oridentical.

A 2-plane in F4 is a set of the form π = {x + t1v1 + t2v2 : t1, t2 ∈ F} wherex, v1, v2 ∈ F4 and v1, v2 are linearly independent. Two lines are coplanar if they liein the same 2-plane; observe that coplanar lines must either be identical, parallel,or intersect in a point. A pair of lines are skew if they are not coplanar.

A 3-space in F4 is a set of the form λ = {x + t1v1 + t2v2 + t3v3 : t1, t2, t3 ∈ F}

where x, v1, v2, v3 ∈ F4 and v1, v2, v3 are linearly independent. Observe that anypair of skew lines lies in a unique 3-space. Two 3-spaces are parallel if they aredisjoint, and one is the translate of the other.

We shall use the symbol p to refer to points, l to lines, π to 2-planes, and λ to3-spaces. We use the symbol P to refer to sets of points, L to sets of lines, 5 tosets of 2-planes, and 3 to sets of 3-spaces. We use Gr(F4, 1) to denote the spaceof all lines, Gr(F4, 2) to denote the space of all 2-planes, and Gr(F4, 3) to denotethe space of all 3-spaces. (Note that these are the affine Grassmannians, in that thespaces do not need to contain the origin).

3. The counterexample

It is likely that Proposition 1.3 follows from the standard theory of Fano varietiesof quadric surfaces, but we will just give an elementary argument.

Proof of Proposition 1.3. Let P and L be as in the proposition. It is clear from theconstruction that the lines in L lie in P . Now we verify the cardinality bounds.We begin with a standard lemma on the number of ways of representing a fieldelement as a quadratic form.


Lemma 3.1. Let 〈 , 〉 : Fn× Fn

→ F be a symmetric bilinear form on Fn withrank at least 1, and let Q(x) := 〈x, x〉 be the associated quadratic form. Then

(3–1) {(x1, . . . , xn) ∈ Fn: Q(x1, . . . , xn)= x} . |F |

n−1

for all x ∈ F.If we know that 〈 , 〉 has rank at least 3, we can improve (3–1) to

(3–2) {(x1, . . . , xn) ∈ Fn: Q(x1, . . . , xn)〉 = x} ∼ |F |

n−1

for all x ∈ F, if |F | is sufficiently large.

Proof. By placing the quadratic form Q in normal form (recalling that charF 6= 2)we may assume that

Q(x1, . . . , xn)= α1x21 + · · · +αk x2

k ,

where k is the rank of Q and α1, . . . , αk are nonzero elements of F . We mayassume that k = n since the general case n ≥ k follows by adding n − k dummyvariables. In particular α j 6= 0 for j = 1, . . . , n.

The bound (3–1) is now clear, since if we fix x1, . . . , xn−1 and x then there areat most 2 choices for xn . Now let us assume n ≥ 3, and prove (3–2).

We use Gauss sums. We fix a nonprincipal character e of F , i.e. a multiplicativefunction e : F → S1 that is not identically 1. For instance, if F = Z/pZ for someprime p, one can take e(x) := exp(2π i x/p).

For any y ∈ F , let S(y) be the Gauss sum S(y) :=∑

x∈F e(yx2). As is wellknown (see [Mockenhaupt and Tao 2004], for example), S(0) is equal to |F |, while|S(y)| = |F |

1/2 for all nonzero values of y.Fix x ∈ F . By expanding the Kronecker delta as a Fourier series, we see that

the number of solutions to (3–1) can be written as∑x1,...,xn∈F

δ(α1x21+ · · · +αnx2

n − x)=1

|F |

∑y∈F

∑x1,...,xn∈F

e((α1x21+ · · · +αnx2

n − x)y)

=1

|F |

∑y∈F

e(−xy)n∏

i=1

S(αi y)

= |F |n−1

+1

|F |

∑y∈F\{0}

e(−xy)n∏

i=1

S(αi y)

= |F |n−1

+1

|F |

∑y∈F\{0}

O(|F |n/2)

= |F |n−1

+ O(|F |n/2)

as desired, since n ≥ 3. �

342 TERENCE TAO

From the lemma we see that |P|∼ |F |3, as desired. Now we count the lines in L .

The lemma yields ∼ |F |3 choices of null direction {v ∈ F4

\0 : 〈v, v〉 = 0}.For each such v, the space v⊥

:= {x ∈ F4: 〈x, v〉 = 0} is 3-dimensional (since

Q is nondegenerate). Furthermore, since Q is nondegenerate on F4 and v is anull vector, we see that Q must also be nondegenerate on v⊥. Restricting Q tov⊥ (which is of course isomorphic to F3) we see from Lemma 3.1 that there are∼ |F |

2 choices for x . Thus there are ∼ |F |5 possible pairs (x, v) that generate a

line in L . However, each line in L is generated by ∼ |F |2 such pairs (x, v), so we

have |L| ∼ |F |3 as desired.

It remains to verify the Wolff axiom. First pick a 3-space λ and consider thelines in L which go through λ.

Pick an arbitrary point x0 in λ, so that λ− x0 is a three-dimensional subspace ofF4. By Lemma 3.1, the number of null vectors {v ∈ λ−x0 : 〈v, v〉 = 0} is O(|F |

2).Fix v as above. There are two cases. If v⊥

6≡ (λ− x0), then there are O(|F |)

choices of x ∈ λ such that 〈x, v〉 = 0 and 〈x, x〉 = 1. But if v⊥≡ (λ− x0), then

the number of choices for x could be as large as O(|F |2). But (λ − x0)

⊥ onlyhas cardinality O(|F |), hence the number of v in the second category is at mostO(|F |). Thus the number of pairs (x, v) which can generate a line in λ is at mostO(|F |

3). But each line is generated by ∼ |F |2 pairs (x, v). Thus the number of

lines in λ is at most O(|F |), which clearly implies the Wolff axiom for both k = 2and k = 3. This completes the proof of Proposition 1.3. �

4. Some basic algebraic geometry

Here we review some basic facts from algebraic geometry (for proofs see [Harris1992], for example), and apply them to our Kakeya problem. The material wewill need is not very advanced; basically, we need the concept of the dimensionof an algebraic variety, and we need to know that this dimension behaves in theexpected way with respect to intersections, projections, cardinality, etc. We shallalso rely heavily on the basic fact that the dimension of an algebraic variety isalways an integer (in contrast to, say, the “half-dimensional” field G mentioned inthe introduction).

Let F denote the algebraic closure of F and n ≥ 1. An algebraic variety in Fn

is defined to be the zero locus of a collection Q1, . . . , Qk of F-valued polynomialson the affine space F

n. In this paper we shall always assume that our algebraic

varieties have bounded degree, thus k = O(1) and all the polynomials Q1, . . . , Qk

have degree O(1).An algebraic variety V in F

nhas a well-defined integer-valued dimension 0 ≤

d ≤ n; there are several equivalent definitions of this dimension, for instance d is


the smallest nonnegative integer such that generic affine spaces in Fn

of codimen-sion greater than d are disjoint from V . (See [Harris 1992] for more equivalentdefinitions of dimension). If V has dimension n then it must be all of F

n, while

if V has dimension 0 then it can only consist of at most O(1) points. Of course,the algebraic geometry notion of dimension is consistent with the linear algebranotion of dimension, thus for instance 3-spaces have dimension 3.

An algebraic variety is irreducible if it does not contain any proper sub-varietyof the same dimension. Every algebraic variety of dimension k can be decomposedas a union of O(1) irreducible varieties of dimension at most k.

We define an algebraic variety in Fn of dimension d to be a restriction to Fn ofan algebraic variety in F

nof dimension d . Observe that if V is a variety in Fn of

dimension d then |V | . |F |d (this can be shown, for instance, by taking generic

intersections with affine spaces of codimension d).Let L ⊆ Gr(Fn, 1) be a collection of lines which point in different directions. In

the introduction we observed that this implies the Wolff axiom, that not too manylines in L can lie inside a k-space. In fact we can generalize this to k-dimensionalvarieties; see [Mockenhaupt and Tao 2004, Proposition 8.1]:

Lemma 4.1 (Generalized Wolff property). Let V ⊆ Fn be an algebraic variety inFn of dimension k, and let L ⊆ Gr(Fn, 1) be a collection of lines in Fn which pointin different directions. Then∣∣{l ∈ L : l ⊆ V }

∣∣. |F |k−1.

Remark. The lines in Proposition 1.3 violate this property, but of course they donot point in different directions. (On the other hand, one can show that the linesarising from the Heisenberg example do obey this generalized Wolff property. Itmay be that in the three-dimensional theory, one needs to extend this lemma further,to cover not only varieties over F , but also over subfields of F such as G.)

Proof. We may of course assume that |F | � 1, since the claim is obvious for |F |

bounded.We can embed Fn in the projective space P Fn+1, which we think of as the union

of Fn with the hyperplane at infinity. By replacing the defining polynomials of Vwith their homogeneous counterparts, we can thus extend V to a k-dimensionalvariety V in P Fn+1 (see e.g. [Harris 1992]).

We break up V into irreducible components, each of dimension at most k. Wecan assume that none of the irreducible components are contained inside the hyper-plane at infinity, since we could simply remove those components and still have anextension of V . In particular we see that the intersection of V with the hyperplaneat infinity is at most k − 1-dimensional.

344 TERENCE TAO

Let l be a line in L , which we can extend to be a projective line l in P Fn+1 byadding a single point at infinity (the direction of l). Observe that the restriction ofV to l is an algebraic variety of dimension either 0 or 1; in other words, either theprojective line l lies inside V , or else l intersects V in at most O(1) points. Thusin order for l to be contained in V , the direction of l must lie inside V (assumingthat |F | is sufficiently large). But by the previous paragraph the number of suchdirections is at most O(|F |

k−1). Since the lines in L point in different directions,we are done. �

As a consequence of this lemma we see that a Besicovitch set cannot have highintersection with an algebraic variety:

Corollary 4.2. Let V ⊆ Fn be an algebraic variety in Fn of dimension at mostn − 1, and let L ⊆ Gr(Fn, 1) be a collection of lines in Fn which point in differentdirections. Then ∣∣{(p, l) ∈ V × L : p ∈ l}

∣∣. |F |n−1.

The trivial upper bound for the left-hand side is |F ||L|. |F |n . Thus this lemma

gains a power of |F | over the trivial bound.

Proof. As in the proof of Lemma 4.1, we observe that every line l in Fn is eithercontained in V , or else intersects V in at most O(1) points. The lines of thesecond type contribute at most O(|L|)= O(|F |

n−1) incidences, while by Lemma4.1 the lines of the first type contribute at most O(|F ||F |

dim(V )−1) = O(|F |n−1)

incidences, and we are done. �

A further consequence is that the lines of a Besicovitch set cannot have largeintersection with an algebraic variety:

Corollary 4.3. Let L ⊆ Gr(Fn, 1) be a collection of lines in Fn which point indifferent directions, and let P ⊆ Fn be a set of points containing all the lines inL. Let W ⊆ Gr(Fn, 1) be an algebraic variety of lines of dimension at most n − 1.Then

|L ∩ W | . |F |n−2

+ |F |−1

|P|.

Again, this lemma gains a power of |F | over the trivial bound of |F |n−1 (as-

suming P is not too huge).

Proof. Consider the set

X := {(p, l) ∈ Fn× W : p ∈ l}.

This is an algebraic variety in Fn×Gr(Fn, 1) of dimension at most n. Now consider

the map φ : X → Fn given by φ(p, l) := p. Observe that for any p in the image of φ,the fibers φ−1(p) are either 0-dimensional (i.e. have cardinality O(1)), or at least1-dimensional. This implies (see e.g. [Harris 1992]) that we have a decomposition


φ(X) := P1 ∪ P2, where the fibers φ−1(p) are 0-dimensional for all p ∈ P1, andP2 is contained in an algebraic variety of dimension at most n − 1.

By the construction of P1 we have

|{(p, l) ∈ P1 × (L ∩ W ) : p ∈ l}

∣∣. |{

p ∈ P1 : p ∈ l for some l ∈ L ∩ W }∣∣

. |{

p ∈ P1 : p ∈ P}∣∣

≤ |P|.

Also, by Corollary 4.2 we have∣∣{(p, l) ∈ P2 × (L ∩ W ) : p ∈ l}∣∣. |F |

n−1.

Adding the two estimates, we see that

|F ||L ∩ W | =∣∣{(p, l) ∈ φ(X)× (L ∩ W ) : p ∈ l}

∣∣. |F |n−1

+ |P|

and the claim follows. �

To apply these results to our four-dimensional problem, we need some notationfor reguli.

Definition 4.4. A frame f is a quadruplet f = (l1, l2, l3, λ) where λ ∈ Gr(F4, 3)is a 3-space in F4, and l1, l2, l3 ∈ Gr(F4, 1) are distinct, mutually skew lines in F4

which lie inside λ. If f is a frame, we write λ( f ) for λ. If f = (l1, l2, l3, λ) is aframe, we use L( f ) to denote the set of lines l ∈ Gr(F4, 1) which intersect l1, l2,and l3, and r( f )⊆ λ to denote the union of all the lines in L( f ).

The set r( f ) is called the regulus generated by the frame f . It is a quadric inλ, that is, the zero locus of a quadratic polynomial in λ, and hence an algebraicvariety of dimension 2. (The prototypical regulus is the set {(x, y, xy, 0) ∈ F4

:

x, y ∈ F}, where the lines li are of the form {(x, yi , xyi , 0) : x ∈ F} for somedistinct y1, y2, y3 ∈ F . All reguli can be shown to be projectively equivalent to thisexample.) Since the lines in a frame are mutually skew, we see that this quadraticpolynomial is irreducible (so the regulus is not a (double) plane, or the union of twoplanes), and that the lines L( f ) have cardinality ∼|F | and are finitely overlapping.

Corollary 4.5 (Three-regulus lemma). Let L ⊆ Gr(F4, 1) be a collection of lines inF4 which point in different directions, and let P ⊆ F4 be a set of points containingall the lines in L. Let f1, f2, f3 be three frames such that the 3-spaces λ( f1), λ( f2),λ( f3) are parallel and disjoint. Then∣∣{l ∈ L : l ∩ r( fi ) 6= ∅ for all i = 1, 2, 3}

∣∣. |F |2+ |F |

−1|P|.

Again, this bound improves by roughly |F | over the trivial bound of |F |3, if |P|

is not much larger than |F |3. The hypothesis that the 3-spaces λ( fi ) are parallel

can be substantially relaxed, but we will not need to do so here.

346 TERENCE TAO

Proof. Fix f1, f2, f3, and let W ⊆ Gr(F4, 1) denote the set

(4–1) W := {l ∈ Gr(F4, 1) : l ∩ r( fi ) 6= ∅ for all i = 1, 2, 3}.

Since the r( fi ) and l are algebraic varieties, it is clear (e.g. by using resultants;see e.g. [Harris 1992]) that the relationship l ∩ r( fi ) 6= ∅ is equivalent to somefinite set of explicit algebraic relations between the defining parameters of l and fi .Thus W is an algebraic variety in Gr(F4, 1). In light of Corollary 4.3, it will sufficeto verify that W has dimension at most 3. (We apologize to algebraic geometrysophisticates for the appalling crudeness of the following argument.)

Let p be a point in r( f1). Let φp be the stereographic projection from λ( f2) toλ( f3), thus φp(x)= y if and only if p, x , y are collinear. W is isomorphic to

{(p, y) ∈ r( f1)× λ( f3) : y ∈ r( f3)∩φp(r( f2))}

(basically because two points determine a line, and because the planes λ( fi ) aredisjoint). In other words, one can think of W as a bundle over r( f1) whose fiberat p is r( f3)∩φp(r( f2)).

Note that φp : λ( f2)→ λ( f3) is an invertible linear map, so that φp(r( f2)) is anirreducible quadric surface in λ( f3). The set r( f3)∩φp(r( f2)) thus has dimensionat most 2, and in fact will have dimension at most 1 unless φp(r( f2)) ≡ r( f3)

(by irreducibility). However, as p varies, the quadric surfaces φp(r( f2)) move bytranslation. Since r( f2) is not a plane, we thus see that there can be at most aone-dimensional family of points p for which φp(r( f2))≡ r( f3).

To summarize, as p varies over the two-dimensional variety r( f1), the fiberr( f3)∩φp(r( f2)) is at most one-dimensional, except possibly for a one-dimensionalfamily of points p for which the fiber is two-dimensional. From this it is clearthat W has dimension at most 3, and we are done. (To compute the dimensionproperly one should work in the algebraically closed field F here. But this causesno difficulty, since the above geometric considerations are valid for all fields ofcharacteristic larger than two.) �

Corollary 4.5 is the analogue of the three lines lemma used in [Schlag 1998].Our strategy will now be to start with a Besicovitch set and construct many framesf and many lines l ∈ L so that r( f ) intersects L , in order to exploit the aboveCorollary. To do this we shall need some basic combinatorial tools, which we nowpause to review.


5. Some basic combinatorics

We shall frequently use the following elementary observation: If B is a finite setand µ : B → R+ is a function such that∑

b∈B

µ(b)≥ X,

then ∑b∈B:µ(b)≥X/2|B|

µ(b)≥ X/2.

We refer to this as a popularity argument, since we are restricting B to the values bwhich are popular in the sense that µ is large. The argument will be used iterativelymany times.

We shall frequently use a version of the Cauchy–Schwarz and Holder inequali-ties:

Lemma 5.1. Let A, B be finite sets, and let ∼ be a relation connecting pairs(a, b) ∈ A × B such that ∣∣{(a, b) ∈ A × B : a ∼ b}

∣∣& X

for some X � |B|. Then∣∣{(a, a′, b) ∈ A × A × B : a 6= a′; a, a′

∼ b}∣∣& X2

|B|

and ∣∣{(a, a′, a′′, b) ∈ A × A × B : a, a′, a′′ distinct; a, a′, a′′∼ b}

∣∣& X3

|B|2

Proof. Define for each b ∈ B, define µ(b) := |{a ∈ A : a ∼ b}|. By hypothesis, wehave ∑

b∈B

µ(b)& X.

In particular, by the popularity argument we have∑b∈B:µ(b)&X/|B|

µ(b)& X.

By hypothesis, we have X/|B| � 1. From this and the foregoing, we obtain∑b∈B:µ(b)&X/|B|

µ(b)(µ(b)− 1)& X (X/|B|)

and ∑b∈B:µ(b)&X/|B|

µ(b)(µ(b)− 1)(µ(b)− 2)& X (X/|B|)(X/|B|).

348 TERENCE TAO

The claims follow. �

A typical application of Lemma 5.1 is the following standard incidence bound:

Corollary 5.2. For an arbitrarily collection P ⊆ Fn of points and L ⊆ Gr(Fn, 1)of lines, we have

(5–1)∣∣{(p, l) ∈ P × L : p ∈ l}

∣∣. |P|1/2

|L| + |P|

Proof. We may of course assume that the left-hand side of (5–1) is � |P|, sincethe claim is trivial otherwise. From Lemma 5.1 we have∣∣{(p, l, l ′) ∈ P × L × L : p ∈ l ∩ l ′; l 6= l ′}

∣∣& |P|−1∣∣{(p, l) ∈ P × L : p ∈ l}

∣∣2.On the other hand, |l ∩ l ′| has cardinality O(1) if l 6= l ′, thus∣∣{(p, l, l ′) ∈ P × L × L : p ∈ l ∩ l ′; l 6= l ′}

∣∣. |L|2.

Combining the two estimates we obtain the result. �

The preceding estimate will be most useful when |L| is small — in particular if|L| = O(|F |). When |L| is large, we have an alternate estimate:

Proposition 5.3 [Mockenhaupt and Tao 2004]. Let the notation be as in Corollary5.2. If we further assume that the lines in L point in different directions, then

(5–2)∣∣{(p, l) ∈ P × L : p ∈ l}

∣∣. |P|1/2

|L|3/4

|F |1/4

+ |P| + |L|.

Proof. A proof is given after [Mockenhaupt and Tao 2004, Proposition 8.6]; theargument there is essentially due to Nets Katz, but the original result of this typedates back to Wolff [1995; 1998b].

For the convenience of the reader we now sketch an informal “probabilistic”derivation of (5–2). Let I denote the set in (5–2). We may assume that |I | �

|P|, |L| since the claim is trivial otherwise.Observe that a randomly chosen point p ∈ P and a randomly chosen line l ∈ L

have a probability |I |/|P||L| of being incident (so that p ∈ l). Thus, given tworandom lines l1, l2 ∈ L and a random point p ∈ P , we expect2 the chance that p isincident to both l1 and l2 is (|I |/|P||L|)2. Since there are |P| possible values forp, the chance that two random lines l1, l2 ∈ L intersect at all is thus heuristically|P|(|I |/|P||L|)2.

As a consequence, the probability that three random lines l1, l2, l3 ∈ L form atriangle is heuristically (|P|(|I |/|P||L|)2)3. (There is the chance that this triangle

2This of course assumes independence of various random events, which is usually not the case.To make the argument rigorous one must use such tools as Lemma 5.1, which can be viewed as astatement that certain events are positively correlated. See [Mockenhaupt and Tao 2004, Proposition8.6] for details.


is degenerate, but the hypothesis |I | � |P|, |L| can be used to show that the prob-ability of this occurring is low). On the other hand, given two intersecting linesl1, l2 ∈ L , there are at most O(|F |) lines l3 ∈ L which can intersect them both, sincewe may apply the Wolff axiom to the 2-plane spanned by l1 and l2. Combiningthese estimates we obtain(

|P|

(|I |

|P||L|

)2)3

. |P|

(|I |

|P||L|

)2 |F |

|L|

and (5–2) follows.3 �

Remark. One only requires the Wolff axiom on L to obtain (5–2). In particularone can easily obtain Theorem 1.2 as a consequence of (5–2). It is likely that onecan generalize Theorem 1.4 to obtain a further improvement to (5–2), but we donot pursue this question here.

6. A heuristic proof of Theorem 1.4

We now give a heuristic explanation for why we can improve upon Theorem 1.2in four dimensions, in the spirit of the probabilistic arguments in Proposition 5.3.In later sections we shall make this heuristic argument rigorous.

Suppose for contradiction we have a family L ⊆ Gr(F4, 1) of lines in differentdirections of cardinality |L| ∼ |F |

3 which are contained in a set P ⊆ F4, also ofcardinality |P| ∼ |F |

3. Arguing as in Proposition 5.3 we see that any two lines inL have a (heuristic) probability ∼ 1/|F | of intersecting.

Also, a random line l ∈ L and a random 3-space λ∈ Gr(F4, 3) have a probability1/|F |

2 of being incident (so that l ⊆ λ). Thus we expect a 3-space λ to contain|L|/|F |

2∼ |F | lines in L .

Now consider the set of all quintuples (l1, l2, l1, l2, l3) ∈ L5 of lines such thatli intersects l j for all i = 1, 2, j = 1, 2, 3. From the above heuristics we see thatthere should be about |L|

5(1/|F |)6 ∼ |F |9 such quintuples. On the other hand, for

generic quintuples (l1, l2, l1, l2, l3) of the above form, the lines l1, l2, l3 must lie ina 3-space λ, and l1, l2 must lie in the regulus generated by the frame (l1, l2, l3, λ).(For this heuristic argument we ignore the possibility that the quintuple could de-generate).

To count the number of possible reguli, observe that there are |L|2∼|F |

6 choicesfor l1, l2, which determines λ. From our previous heuristic we see that λ cancontain at most O(|F |) choices for l3, thus there are at most O(|F |

7) reguli.

3It is an instructive exercise to obtain similar heuristic probabilistic derivations of such estimatesas (5–1) (using the fact that two random lines intersect in at most one point) or (1–2) (using the factthat a random regulus contains at most O(|F |) lines). See also Section 6 below.

350 TERENCE TAO

Dividing |F |9 by |F |

7, we thus see that a generic regulus r( f ) of the above typemust contain at least |F |

2 pairs (l1, l2) of lines in L . But r( f ) only has O(|F |) linesto begin with. Thus a generic regulus r( f ) must have extremely large intersectionwith P , so that |r( f )∩ P| ∼ |r( f )| ∼ |F |

2.Since a random p ∈ P and l ∈ L have a probability 1/|F |

2 of being incident,this means that a random line l ∈ L and a random regulus r( f ) have a probability∼ 1 of intersecting. In particular, if we select three parallel reguli r( f1), r( f2),r( f3), a large fraction of lines in L must be incident to all three reguli. But thiscontradicts Corollary 4.5, since |L| ∼ |F |

3 and |P| ∼ |F |3.

7. Preliminary refinements

We now begin the rigorous proof of Theorem 1.4, which will broadly follow theheuristic outline of the previous section.

Let P0 ⊆ F4 be a Besicovitch set. We may assume that

(7–1) |P0| / |F |3+

116 .

since the claim is trivial otherwise. We may also assume that |F | � 1 for similarreasons.

Since P0 is a Besicovitch set, there exists a set L0 ⊆ Gr(F4, 1) of lines indifferent directions such that |L0| ∼ |F |

3 and P0 contains every line in L0. Inparticular the incidence set

I0 := {(p, l) ∈ P0 × L0 : p ∈ l}

has cardinality |I0| = |F ||L0| ∼ |F |4.

Given any line l in L0 and a randomly selected 3-space λ in Gr(F4, 3), theprobability that l lies in λ is ∼ 1/|F |

2. Since |L0| ∼ |F |3, one thus expects every

3-space λ contains about |F | lines in L0 on the average. A similar heuristic leadsus to expect every 2-plane π ∈ Gr(F4, 2) to contain at most O(1) lines on theaverage.

Although these statements need not be true for all 3-spaces λ, certain variantsdo hold if we refine L0 and P0 slightly, as stated in the next result. We write A ≈ Bto mean A ' B and A / B.

Proposition 7.1. There exists a quantity

(7–2) 1 . α / N116 ,

a subset 4 P1 of P0 and a subset L1 of L0 such that the following properties hold.

4In the course of this argument we shall need to refine the set P0 to a slightly smaller set P1,and then further to P2, and similarly refine L0 to L1 and then L2, while also refining some auxiliarysets H0 to H1, and F0 to F1 to F2 to F3. These refinements are largely technical and as a first


Many incidences: We have the incidence bound

(7–3)∣∣{(p, l) ∈ P1 × L1 : p ∈ l}

∣∣' |F |4.

Cardinality and multiplicity bounds: We have the cardinality bound

(7–4) |P1| / α|F |3

and the multiplicity bound

(7–5)∣∣{l ∈ L1 : p ∈ l}

∣∣≈ α−1|F |

for all p ∈ P1.

No 3-space degeneracy: For any 3-space λ ∈ Gr(F4, 3), we have

(7–6)∣∣{l ∈ L1 : l ⊂ λ}

∣∣/ α2|F |.

No 2-plane degeneracy: For any 2-plane π ∈ Gr(F4, 2), we have

(7–7)∣∣{l ∈ L1 : l ⊂ π}

∣∣/ α4.

The quantity α measures the improvement over Wolff’s bound |P0| & |F |3. As

one can see from (7–2), it is rather close to 1.

Proof. We follow standard “iterated refinement” arguments (see [Wolff 1998a;Łaba and Tao 2001b; Christ 1998; Tao and Wright 2003]; our argument here isparticularly close to that in [Łaba and Tao 2001b]). The purpose of the iteration ismainly to obtain the property (7–7).

Define the multiplicity function µ0 on P0 by

µ0(p) := |{l ∈ L0 : p ∈ l}|.

Then we have ∑p∈P0

µ0(p)= |I0|.

If we divide µ0(p) into dyadic “pigeonholes” and apply the dyadic pigeonholeprinciple (observing that log |F | ≈ 1), we conclude that there exists a multiplicityα−1

|F | such that ∑p∈P0:µ0(p)∼α−1|F |

µ0(p)≈ |I0| ≈ |F |4.

Fix this α, and define

P ′

0 := {p ∈ P0 : µ0(p)∼ α−1|F |}

approximation one can view these sets as being essentially the same (although the sets F2, F3 aresignificantly smaller than F0, F1).

352 TERENCE TAO

andI ′

0 := {(p, l) ∈ P ′

0 × L0 : p ∈ l} ⊆ I0.

Then by construction, |I ′

0| ' |I0| ∼ |F |4, and

|P ′

0| ≈ |I0|/(α−1

|F |)≈ α|F |3.

By (7–1) we thus have α / N1

16 . To get the other half of (7–2), we observe fromProposition 5.3 that

|I ′

0| . |P ′

0|1/2

|L0|3/4

|F |1/4

+ |P ′

0| + |L0|;

applying the above estimates, we thus obtain α & 1. Thus (7–2) holds.Set N := log log |F |; the point of this choice of N is that both |F |

C/N and C N

are ≈ 1 for any fixed choice of constant C . We shall inductively construct sets

(7–8) P ′

0 =: P (0) ⊃ P (1) ⊃ · · · ⊃ P (N )

and

(7–9) L0 =: L(0) ⊃ L(1) ⊃ · · · ⊃ L(N )

as follows.5

As indicated above, we set P (0) := P ′

0 and L(0) := L0. Now suppose inductivelythat P (k) and L(k) have already been constructed for some 0 ≤ k < N . We definethe incidence set

I (k) := {(p, l) ∈ P (k) × L(k) : p ∈ l}.

Clearly we have ∑l∈L(k)

|l ∩ P (k)| = |I (k)|.

Thus if we set

L(k+1):=

{l ∈ L(k) : |l ∩ P (k)| ≥

|I (k)|2|L(k)|

}then by the popularity argument we have∑

l∈L(k+1)

|l ∩ P (k)| ≥ |I (k)|/2.

We rewrite this as ∑p∈P(k)

∣∣{l ∈ L(k+1): p ∈ l}

∣∣≥ |I (k)|/2.

5The use of such a large number of refinements is of course overkill (one could probably get awaywith N = 5, in fact), but reducing the number of refinements used does not alter the exponent 1

16 ,since FC/N and C N were ≈ 1 anyway.


Thus if we set

P (k+1):=

{p ∈ P (k) :

∣∣{l ∈ L(k+1): p ∈ l}

∣∣≥ |I (k)|4|P (k)|

},

we get, again by the popularity argument,∑p∈P(k+1)

∣∣{l ∈ L(k+1): p ∈ l}

∣∣≥ |I (k)|/4

or in other words|I (k+1)

| ≥ |I (k)|/4.

We repeat this construction for k = 0, 1, . . . , N − 1, creating a nested sequenceof sets of points (7–8) and sets of lines (7–9). By construction and the fact that4−N

≈ 1, we clearly have

|I (k)| ≈ |I (0)| = |I ′

0| ' |F |4

for all k. Furthermore,

|P (k)| ≤ |P ′

0| / α|F |3 and |L(k)| ≤ |L0| . |F |

3.

Thus, setting P1 := P (N ) and L1 := L(N−1), we see that (7–3), (7–4), and (7–5)hold. (To get the upper bound in (7–5), simply bound the left-hand side by µ0(p).)

It remains only to verify the nondegeneracy conditions (7–6), (7–7).We first verify (7–6). Let λ be a 3-space. Since λ is clearly an algebraic variety

of dimension 3, we can invoke Corollary 4.2 and conclude that∣∣{(p, l) ∈ λ× L(N−2): p ∈ l}

∣∣. |F |3.

From the construction of P (N−1) we thus have

|P(λ)| / |F |2α

where P(λ) := λ∩ P (N−1).Let L(λ) denote those lines in L1 which lie in λ. By the construction of L1 we

have ∣∣{(p, l) ∈ P(λ)× L(λ) : p ∈ l}∣∣' |F ||L(λ)|.

On the other hand, from Proposition 5.3 we have∣∣{(p, l) ∈ P(λ)× L(λ) : p ∈ l}∣∣/ |P(λ)|1/2|L(λ)|3/4|F |

1/4+ |P(λ)| + |L(λ)|.

Combining all three estimates and using (7–2) we obtain

|L(λ)| / α2|F |

which is (7–6).In fact, this argument gives (7–6) if L1 is replaced by L(k) for any 1 ≤ k ≤ N −1.

354 TERENCE TAO

We now prove (7–7). Following [Wolff 1998a; Łaba and Tao 2001b], we definethe plate number pk for 0 ≤ k ≤ N − 1 to be the quantity

pk := supπ∈Gr(F4,2)

∣∣{l ∈ Lk : l ⊂ π}∣∣.

We observe the bounds

(7–10) 1 ≤ pk . |F |;

the former bound comes since Lk is nonempty, while the latter bound comes sincea 2-plane can contain at most O(|F |) lines in different directions.

Clearly the plate numbers are nonincreasing in k. From this, (7–10), the pigeon-hole principle and the fact that |F |

1/N≈ 1, we can find 2 ≤ k ≤ N − 1 such that

(7–11) pk−1 ≈ pk .

Fix this k. We can find a 2-plane π ∈ Gr(F4, 2) such that the set

Lk(π) := {l ∈ Lk : l ⊂ π}

has cardinality pk .Fix π , and let Pk(π) denote the set

Pk(π) := Pk ∩π.

By the construction of Lk , every line in Lk contains ' |F | points in Pk , thus everyline in Lk(π) contains ' |F | points in Pk(π). In particular we see that∣∣{(p, l) ∈ Pk(π)× Lk(π) : p ∈ l}

∣∣' |F | pk .

Applying Corollary 5.2 we conclude that

|F | pk / |Pk(π)|1/2 pk + |Pk(π)|;

from this and (7–10) we thus have

(7–12) |Pk(π)| ' |F | pk .

Let P ′

k(π) denote the space of all points p in Pk(π) such that at least half of allthe lines in {l ∈ Lk−1 : p ∈ l} are contained in Lk−1(π). We have two cases.

Case 1 (parallel case): |P ′

k(π)| ≥12 |Pk(π)|. In this case we have∣∣{(p, l) ∈ Pk(π)× Lk−1(π) : p ∈ l}

∣∣≥ ∣∣{(p, l) ∈ P ′

k(π)× Lk−1(π) : p ∈ l}∣∣

≥12

∣∣{(p, l) ∈ P ′

k(π)× Lk−1 : p ∈ l}∣∣

' |P ′

k(π)|α−1

|F |

' |Pk(π)|α−1

|F |,


while by definition of pk−1 we have

|Lk−1(π)| ≤ pk−1.

Applying Corollary 5.2 we thus see that

|Pk(π)|α−1

|F | / |Pk(π)|1/2 pk−1 + |Pk(π)|,

which by (7–2) implies that

|Pk(π)||F |2 / α2 p2

k−1.

But combining this with (7–11), (7–12) we obtain

pk ' α−2|F |

3.

But this contradicts (7–10) by (7–2). Hence this case cannot occur.

Case 2 (transverse case): |P ′

k(π)| ≤12 |Pk(π)|. In this case we have (by a compu-

tation similar to Case 1)∣∣{(p, l) ∈ Pk(π)× Lk : p ∈ l; l 6⊂ π}∣∣' α−1

|Pk(π)||F |.

Thus, if L∗

k−1 denotes the lines l ∈ Lk−1 which are incident to a point in Pk(π) butare not contained in π , then we have

(7–13) |L∗

k−1| ' α−1|Pk(π)||F | ' α−1

|F |2 pk

by (7–12).We now use Wolff’s hairbrush argument [Wolff 1995], [Wolff 1998b], as mod-

ified to deal with plates in [Wolff 1998a], [Łaba and Tao 2001b]. We can foliateL∗

k−1 as the disjoint union of

L∗

k−1(λ) := {l ∈ L∗

k−1 : l ∈ λ}

where λ ranges over the 3-spaces containing π . For each such λ, observe from theanalogue of (7–6) for L(k−1) that

(7–14) |L∗

k−1(λ)| / α2|F |.

Also, if we defineP∗

k−1(λ) := {p ∈ Pk−1 : p ∈ λ\π}

then by the construction of Lk−1, we have∣∣{(p, l) ∈ P∗

k−1(λ)× L∗

k−1(λ) : p ∈ l}∣∣' |L∗

k−1(λ)||F |.

Applying Corollary 5.2 we obtain

|L∗

k−1(λ)||F | / |P∗

k−1(λ)|1/2

|L∗

k−1(λ)| + |P∗

k−1(λ)|,

356 TERENCE TAO

which by (7–14), (7–2) implies that

|P∗

k−1(λ)| ' α−2|L∗

k−1(λ)||F |.

Summing in λ, we obtain

|Pk−1| ' α−2|L∗

k−1||F | ' α−3|F |

3 pk .

Since |Pk−1| / α|F |3 by construction, we obtain pk / α4, and the claim follows.

�

8. Construction of reguli

We now continue the proof of Theorem 1.4. We begin by refining P1 and L1 alittle further. By (7–3) we have∑

l∈L1

|l ∩ P1| ≈ |F |4.

Thus if we setL2 :=

{l ∈ L1 : |l ∩ P1| ≈ |F |

}then by the popularity argument we get∑

l∈L2

|l ∩ P1| ≈ |F |4

or equivalently ∑p∈P1

∣∣{l ∈ L2 : p ∈ l}∣∣≈ |F |

4.

Thus if we set

P2 :={

p ∈ P1 : |{l ∈ L2 : p ∈ l}| ≈ α−1|F |}

then by (7–4), (7–5), and the popularity argument we have

(8–1)∑p∈P2

∣∣{l ∈ L2 : p ∈ l}∣∣≈ |F |

4.

In particular,

(8–2) |P2| ≈ α|F |3.

The next task is to generate a large number of frames, and a large number oflines in L incident to the reguli generated by these frames. As a frame is a fairlycomplicated combinatorial object (consisting of three lines and a 3-space), we willfirst begin by counting some simpler objects which eventually will be combinedtogether to form frames.


By (8–1) we have ∣∣{(p, l) ∈ P2 × L2 : p ∈ l}∣∣≈ |F |

4.

Since |L2| / |F |3, we thus see from Lemma 5.1 that∣∣{(p1, p2, l) ∈ P2 × P2 × L2 : p1, p2 ∈ l; p1 6= p2}

∣∣' |F |8/|L2| ≈ |F |

5.

By the definition of P2, we see that for each (p1, p2, l) as above there are 'α−1|F |

lines l1 ∈ L2 that contain p1 but are distinct from l, and similarly there are 'α−1|F |

lines l2 ∈ L2 containing p2 but distinct from l. Thus

|H0| ' α−2|F |

7,

where

H0 :={(p1, p2, l, l1, l2) ∈ P2 × P2 × L2 × L2 × L2 :

p1, p2 ∈ l; p1 6= p2; p1 ∈ l1; p2 ∈ l2; l 6= l1, l2}

is the space of “H-shaped” objects.Let H1 ⊆ H0 be the set of elements (p1, p2, l, l1, l2) in H0 such that l1 and l2

are skew. We claim that|H0\H1| / α−1

|F |6.

Indeed, to choose an element (p1, p2, l, l1, l2) in H0\H1 (which is a degenerate H,i.e. a triangle), we first choose p1 ∈ P2 (of which there are / α|F |

3 choices), andthen choose the distinct lines l, l1 incident to p1 (of which there are / (α−1

|F |)2

choices). Since l2 must lie in the 2-plane generated by l and l1, and the lines ofL1 point in different directions, there are only O(|F |) choices for l2. Since p2 isuniquely determined as p2 = l ∩ l2, the claim follows.

From the bounds above and (7–2) we see that

(8–3) |H1| ' α−2|F |

7.

By construction, if h = (p1, p2, l, l1, l2) ∈ H1, then l1 and l2 are skew. Thus l1 andl2 lie in a unique 3-space λ(h), which then must also contain p1, p2, l.

Let S0 ⊂ L2 × L2 denote the pairs (l1, l2) of skew lines in L2. For each pair(l1, l2) ∈ S0, we define the connecting set C(l1, l2) ⊂ L2 to be the set of all linesl ∈ L2 which are distinct from l1, l2, but intersect both l1, l2 in points p1 ∈ P2 andp2 ∈ P2 respectively. Observe the identity∑

(l1,l2)∈S0

|C(l1, l2)| = |H1|.

Since |S0| ≤ |L2|2 . |F |

6, we thus see from (8–3) that if we define

S1 :={(l1, l2) ∈ S0 : |C(l1, l2)| ' α−2

|F |},

358 TERENCE TAO

the popularity argument yields

(8–4)∑

(l1,l2)∈S1

|C(l1, l2)| ' α−2|F |

7.

If (l1, l2) ∈ S1, we define the set C (3)(l1, l2) ⊆ C(l1, l2)3 to be the space of all

triplets (l1, l2, l3)∈ C(l1, l2)3 such that the six points l i

∩l j for i = 1, 2, 3, j = 1, 2are all disjoint.

We now use the nondegeneracy property (7–7) to obtain a lower bound for thesize of C (3)(l1, l2).

Lemma 8.1 (Many triple connections between skew lines). For any (l1, l2) ∈ S1,we have |C (3)(l1, l2)| ' α−4

|F |2|C(l1, l2)|.

Proof. Fix l1, l2. We choose l1∈ C(l1, l2) arbitrarily; of course, there are |C(l1, l2)|

choices for l1.Fix l1. From (7–7) we have∣∣{l2

∈ C(l1, l2) : l2∩ l1 = l1

∩ l1}∣∣/ α4

(since such lines lie in the 2-plane spanned by l1∩ l1 and l2. Similarly if the roles

of l1 and l2 are interchanged. Since

|C(l1, l2)| ' α−2|F |,

we thus see from (7–2) that there are ' α−2|F | choices for l2 such that

l2∩ l j 6= l1

∩ l j for j = 1, 2.

Fix l2. Arguing as above we see that there are ' α−2|F | choices for l3 such that

l3∩ l j 6= l i

∩ l j for i = 1, 2 and j = 1, 2. The claim follows. �

From this lemma and (8–4) we see that∑(l1,l2)∈S1

|C (3)(l1, l2)| ' α−6|F |

9.

Observe that if (l1, l2) ∈ S1 and (l1, l2, l3) ∈ C (3)(l1, l2), the various incidenceassumptions in the definition of S1 and C (3)(l1, l2) force f := (l1, l2, l3, λ) to bea frame, where λ is the unique 3-space spanned by l1 and l2. Observe that l1, l2

both lie in L2 ∩ L( f ). Thus, if F0 denotes the space of all frames generated in thismanner, then

(8–5)∑f ∈F0

|L2 ∩ L( f )|2 ' α−6|F |

9.

Let f ∈ F0. Since the lines in L( f ) are contained in a regulus, they have finiteoverlap. Since each line in L2 contains ≈ |F | points in P1 by construction, we thus


see that6

|P1 ∩ r( f )| & |F | |L2 ∩ L( f )|

so by (8–5) we have ∑f ∈F0

|P1 ∩ r( f )|2 ' α−6|F |

11.

By (7–5), each point in P1 is incident to ≈ α−1|F | lines in L1. Thus we have∑

f ∈F0

∣∣{l ∈ L1 : l ∩ r( f )∩ P1 6= ∅}∣∣2 ' α−8

|F |13.

We observe the cardinality bound

(8–6) |F0| / α2|F |

7.

Indeed, to choose a frame (l1, l2, l3, λ) in F0, we observe that there are

O(|L1|2)= O(|F |

6)

choices for the skew pair (l1, l2). This determines λ, and then by (7–6) we thussee that there are O(α2

|F |) choices for l3, and (8–6) follows. In particular, if wedefine

(8–7) F1 := { f ∈ F0 :∣∣{l ∈ L1 : l ∩ r( f )∩ P1 6= ∅}

∣∣' α−5|F |

3}

then by the popularity argument

(8–8)∑f ∈F1

∣∣{l ∈ L1 : l ∩ r( f )∩ P1 6= ∅}∣∣2 ' α−8

|F |13.

Since the summand on the left-hand side can be crudely bounded by |L1|2

=

O(|F |6), we thus have the crude bound 7

(8–9) |F1| ' α−8|F |

7

(compare with (8–6)).For any frame f ∈ F1, there are only O(|F |

3) possible orientations for λ( f ). By(8–9) and the pigeonhole principle, there therefore exists a 3-space λ0 ∈ Gr(F4, 3)such that

(8–10) |F2| ' α−8|F |

4

6One could also obtain this bound using Corollary 5.2 and the crude bound

|L2 ∩ L( f )| ≤ |L( f )| . |F |.7The bounds on |F1|, and later on |F2|, |F3|, might not be best possible, however an improve-

ment on this part of the argument does not directly improve the gain 116 .

360 TERENCE TAO

whereF2 := { f ∈ F1 : λ( f ) is a translate of λ0}.

Fix this λ0. Let F3 be a maximal subset of F2 such that the reguli {r( f ) : f ∈ F3}

are all distinct. Since each r( f ) contains at most O(|F |) lines, each regulus canarise from at most O(|F |

3) frames. We thus see from (8–10) that

(8–11) |F3| ' α−8|F |.

From (8–7) we have∑f ∈F3

∣∣{l ∈ L1 : l ∩ r( f )∩ P1 6= ∅}∣∣' α−5

|F |3|F3|.

From (7–2), (8–11) the right-hand side is � |F |3 & |L1|. Thus we can use Lemma

5.1, and obtain∑f1, f2, f3∈F3

f1, f2, f3 distinct

∣∣{l ∈ L1 : l ∩ r( fi )∩ P1 6= ∅ for i = 1, 2, 3}∣∣' α−15

|F |3|F3|

3.

From the pigeonhole principle, we may thus find distinct frames f1, f2, f3 in F3

such that

(8–12) |L∗| ' α−15|F |

3,

where L∗ ⊆ L1 is the collection of lines

L∗ := {l ∈ L1 : l ∩ r( fi )∩ P1 6= ∅ for i = 1, 2, 3}.

Now we consider the problem of obtaining upper bounds on |L∗|. The crude up-per bound of |L1| ∼ |F |

3 is clearly not enough to obtain a contradiction. However,thanks to the three-regulus lemma we can improve this bound by about |F |:

Proposition 8.2. We have

(8–13) |L∗| / |F |2+

116 .

Proof. If the 3-spaces λ( f1), λ( f2), λ( f3) are disjoint, this follows directly fromCorollary 4.5 and (7–1).

By symmetry it remains to consider the case when λ( f1) and λ( f2) (for instance)are equal. Then the lines in L∗ must either be parallel to λ( f1), or else intersectP1∩r( f1)∩r( f2). There are at most |F |

2 lines in the first category (in fact there arefar fewer, thanks to (7–6)). In the second category, we observe that r( f1)∩r( f2) isat most one-dimensional (since r( f1), r( f2) are irreducible and distinct) and hencehas cardinality O(|F |). On the other hand, by (7–5) every point in r( f1)∩r( f2)∩P1

is incident to ≈ α−1|F | lines in L1. Thus we certainly have / |F |

2+116 incidences

in this case as well. �


Combining (8–13) with (8–12) we obtain

α ' |F |1

16

and hence by (8–2)|P0| ' |P2| ≈ α|F |

3 ' |F |3+

116

as desired. This concludes the proof of Theorem 1.4.

9. Remarks

It seems likely that this Theorem can be generalized in several ways. The expo-nent 1

16 is probably not sharp, and also the result should have extension to otherdimensions, perhaps through more sophisticated use of algebraic geometry. In di-mensions 5 and higher there are other, more “arithmetic” arguments that give slightimprovements to |F |

(n+2)/2 for Besicovitch sets; see [Bourgain 1999; Katz and Tao1999; 2002a; 2002b; Mockenhaupt and Tao 2004; Rogers 2001]. Nevertheless, ifone can make an improvement of the order of 1

16 in, say, five dimensions by these“geometric” techniques, this will be quite competitive with the results in, say, [Katzand Tao 2002b]. In the Euclidean setting one can improve the bound (n + 2)/2 inall dimensions n ≥3 by a small number (10−10) for the upper Minkowski dimensionproblem for Besicovitch sets [Katz et al. 2000; Łaba and Tao 2001a; 2001b], butthis argument seems special to the upper Minkowski problem and does not directlyimpact the finite field question.

Also, the argument can probably be extended to obtain an estimate on theKakeya maximal function for finite fields; see [Mockenhaupt and Tao 2004]. Inprinciple, the finite field results should also extend to the Euclidean setting Rn ,but there are unpleasant technical difficulties in the process, due, for instance, tothe presence of near-degenerate reguli in Rn . Also in the finite field case one isaided considerably by the fact that dimensions must be integer; for instance, theintersection of two lines is either empty, 0-dimensional (a point), or 1-dimensional(a line). In the (δ-discretized) Euclidean case there is a continuum of cases: twodistinct 1 × δ tubes can intersect in a set of length ∼ δ/θ , where δ < θ < 1 is theangle between the two tubes. This introduces a new dyadic parameter θ into theanalysis (measuring the degeneracy of the angle), and often the cases of small θ andlarge θ need to be treated separately. See [Wolff 1995], for example. Here we havemore complicated algebraic objects, such as the variety (4–1), and to capture thepossible degeneracies of this object seems to require a large number of additionaldyadic parameters. It is possible that various rescaling arguments, such as the two-ends and bilinear reductions mentioned above, may be used to reduce the numberof such parameters, but the extension of this argument to the Euclidean case stillappears to be quite nontrivial.

362 TERENCE TAO

Difficulties of these kinds cause considerable complication in such papers as[Schlag 1998], although some could perhaps be alleviated using the “two-ends”reduction in [Wolff 1995] and the “bilinear reduction” in [Tao et al. 1998].

Acknowledgments

The author thanks Tony Carbery, Nets Katz, Wilhelm Schlag, and Jim Wright forhelpful discussions, and is also indebted to David Gieseker and Allen Knutsonfor their explanation of some of the basics of algebraic geometry. The author isparticularly indebted to Nets Katz for emphasizing the importance of reguli to thisproblem. Finally, the author thanks the anonymous referee for a careful readingand the detection of several misprints.

References

[Bourgain 1991] J. Bourgain, “Besicovitch type maximal operators and applications to Fourier anal-ysis”, Geom. Funct. Anal. 1:2 (1991), 147–187. MR 92g:42010 Zbl 0756.42014

[Bourgain 1999] J. Bourgain, “On the dimension of Kakeya sets and related maximal inequalities”,Geom. Funct. Anal. 9:2 (1999), 256–282. MR 2000b:42013 Zbl 0930.43005

[Christ 1998] M. Christ, “Convolution, curvature, and combinatorics: a case study”, Internat. Math.Res. Notices 19 (1998), 1033–1048. MR 2000a:42026 Zbl 0927.42008

[Gowers 1998] W. T. Gowers, “A new proof of Szemerédi’s theorem for arithmetic progressions oflength four”, Geom. Funct. Anal. 8:3 (1998), 529–551. MR 2000d:11019 Zbl 0907.11005

[Harris 1992] J. Harris, Algebraic geometry, Graduate Texts in Mathematics 133, Springer, NewYork, 1992. MR 93j:14001 Zbl 0779.14001

[Katz and Tao 1999] N. H. Katz and T. Tao, “Bounds on arithmetic projections, and applications tothe Kakeya conjecture”, Math. Res. Lett. 6:6 (1999), 625–630. MR 2000m:28006 Zbl 0980.42013

[Katz and Tao 2002a] N. Katz and T. Tao, “Recent progress on the Kakeya conjecture”, Publ. Mat.Vol. Extra (2002), 161–179. MR 2003m:42036 Zbl 1024.42010

[Katz and Tao 2002b] N. H. Katz and T. Tao, “New bounds for Kakeya problems”, J. Anal. Math.87 (2002), 231–263. MR 2003i:28006 Zbl 1027.42014

[Katz et al. 2000] N. H. Katz, I. Łaba, and T. Tao, “An improved bound on the Minkowski di-mension of Besicovitch sets in R3”, Ann. of Math. (2) 152:2 (2000), 383–446. MR 2002i:28006Zbl 0980.42014

[Łaba and Tao 2001a] I. Łaba and T. Tao, “An improved bound for the Minkowski dimension of Be-sicovitch sets in medium dimension”, Geom. Funct. Anal. 11:4 (2001), 773–806. MR 2003b:28006Zbl 1005.42009

[Łaba and Tao 2001b] I. Łaba and T. Tao, “An x-ray transform estimate in Rn”, Rev. Mat. Ibero-americana 17:2 (2001), 375–407. MR 2003a:44003 Zbl 1024.44002

[Mockenhaupt and Tao 2004] G. Mockenhaupt and T. Tao, “Restriction and Kakeya phenomena forfinite fields”, Duke Math. J. 121:1 (2004), 35–74. MR 2004m:11200 Zbl 02103577

[Rogers 2001] K. M. Rogers, “The finite field Kakeya problem”, Amer. Math. Monthly 108:8 (2001),756–759. MR 2002g:11175 Zbl 1028.43007

[Schlag 1998] W. Schlag, “A geometric inequality with applications to the Kakeya problem in threedimensions”, Geom. Funct. Anal. 8:3 (1998), 606–625. MR 99g:42025 Zbl 0939.42012


[Tao and Wright 2003] T. Tao and J. Wright, “L p improving bounds for averages along curves”, J.Amer. Math. Soc. 16:3 (2003), 605–638. MR 2004j:42005 Zbl 01896622

[Tao et al. 1998] T. Tao, A. Vargas, and L. Vega, “A bilinear approach to the restriction and Kakeyaconjectures”, J. Amer. Math. Soc. 11:4 (1998), 967–1000. MR 99f:42026 Zbl 0924.42008

[Wolff 1995] T. Wolff, “An improved bound for Kakeya type maximal functions”, Rev. Mat. Ibero-americana 11:3 (1995), 651–674. MR 96m:42034 Zbl 0848.42015

[Wolff 1998a] T. Wolff, “A mixed norm estimate for the X-ray transform”, Rev. Mat. Iberoamericana14:3 (1998), 561–600. MR 2000j:44006 Zbl 0927.44002

[Wolff 1998b] T. Wolff, “Recent work connected with the Kakeya problem”, pp. 129–162 in Pros-pects in mathematics (Princeton, NJ, 1996), edited by H. Rossi, Amer. Math. Soc., Providence, RI,1998. MR 2000d:42010 Zbl 0934.42014

Received April 19, 2002. Revised September 10, 2002.

TERENCE TAO


UCLALOS ANGELES CA 90095-1555

[email protected]://www.math.ucla.edu/˜tao


DUALITY IN EQUIVARIANT KK -THEORY

KLAUS THOMSEN

Let A and B be separable C∗-algebras with actions of a locally compactsecond countable group by automorphisms. We construct a C∗-algebra,Aπ , such that the equivariant KK -groups, KK ∗

G(A, B), of Kasparov is iso-morphic to the K -theory groups of Aπ .

1. Introduction

Duality results for KK -theory started with the work of W. Paschke [1981], who ob-tained a description of the BDF-extension group Ext−1(A) (for which see [Brownet al. 1977]) as the K0-group of the C∗-algebra

π(A)′ ∩ Q = {q ∈ Q : qπ(a)= π(a)q, a ∈ A} ,

where Q is the Calkin algebra and π : A → Q is a certain “large” ∗-homomorphism.This showed that K -homology can be described by K -theory, which is sometimesthought of as the dual of K -homology. The work of Paschke has been general-ized by others; see [Valette 1983; Skandalis 1988; Higson 1995; Thomsen 2001].The general duality result obtained in this last reference formed the basis for theprogress on the calculation of the KK -theory and the E-theory of amalgamated freeproducts obtained in [Thomsen 2003]. Thus sufficiently general results of this kindcan be very useful, and need not be justified solely by their great theoretical appeal.It is the purpose of the present paper to obtain duality results for the equivariantKK -theory of Kasparov [1988]. Specifically, we shall show that the notion of anabsorbing ∗-homomorphism, which is the key to all the above-mentioned dualityresults, makes perfect sense in the equivariant setting and that there always exist(sufficiently nice) absorbing ∗-homomorphisms, also in this case. As a result weare able to associate to any pair of separable G-algebras, A and B, a C∗-algebraAGπ such that

K0(AGπ

)= KK 1

G(A, B)and

K1(AGπ

)= KK 0

G(A, B).

MSC2000: 19K35.Keywords: equivariant K K − theor y, C∗-algebras.

365

366 KLAUS THOMSEN

As the notation should suggest, AGπ is the fixed point algebra of a C∗-algebra

Aπ with an action of G by automorphisms. The π is here a ∗-homomorphismwhich is absorbing in an appropriate way, and it can be chosen such that thecanonical forgetful maps KK i

G (A, B)→ KK i (A, B) , i = 0, 1, become the mapsKi(AGπ

)→ Ki (Aπ ) , i = 0, 1, induced by the embedding AG

π ⊆ Aπ .

2. G-algebras and Hilbert G-modules

This section introduces the basic definitions, and sets up notation and terminologyby describing some fundamental results on Hilbert modules over G-algebras thatare crucial for the following. They are all more or less known, and we omit theproofs. The main result, Theorem 2.8, which is due to R. Meyer, is the cornerstonefor the results of the paper; it gives us access to genuinely equivariant stabilizationresults for Hilbert bimodules over G-algebras, provided the algebra acting fromthe left has been suitably stabilized.

When B is a C∗-algebra and E, F are Hilbert B-modules (see [Kasparov 1980a;1980b; 1988; Lance 1995; Jensen and Thomsen 1991]), we let LB(E, F) denotethe Banach space of adjointable maps from E to F , and by KB(E, F) the idealin LB(E, F) consisting of the “compact” operators, i.e. KB(E, F) is the closedsubspace generated by {θx,y : x ∈ F, y ∈ E}, where θx,y(z) = x〈y, z〉. WhenE = F , LB(E, F) is a C∗-algebra which we denote by LB(E). Similarly, the idealKB(E, F) is denoted by KB(E) in this case. Moreover, when E is the HilbertB-module B itself, we will write M(B) for the multiplier algebra M(B)= LB(B)and B for KB(B).

Let G be a locally compact second countable group.

Definition 2.1. A G-algebra is a pair (A, α) where A is a σ -unital C∗-algebra andα : G → Aut A is a homomorphism such that G 3 g 7→ αg(a) is norm-continuousfor all a ∈ A.

In the following we shall often drop the explicit reference to α and denote theG-algebra (A, α) simply by A. We write then g · a for αg(a). By a C∗-algebrawe mean in the following a G-algebra for which the G-action is trivial. Given twoG-algebras, A and B, the minimal tensor product A⊗ B will be considered, unlessexplicitly stated otherwise, as a G-algebra with what is usually referred to as “thediagonal action”: g · (a ⊗ b)= g · a ⊗ g · b.

Definition 2.2. Let (B, β) be a G-algebra. A Hilbert B,G-module is a pair (E, v),where E be a Hilbert B-module and v is a representation of G as operators on Esuch that the map G × E 3 (g, e) 7→ vg(e) is continuous, and

〈vg(e), vg( f )〉 = βg(〈e, f 〉),(2–1)

for e, f ∈ E , and g ∈ G.

DUALITY IN EQUIVARIANT KK -THEORY 367

Although the operators vt are not adjointable in general, they do give rise to arepresentation of G as automorphisms of LB(E) since vgmvg−1 is adjointable whenm is, and (vgmvg−1)∗ = vgm∗vg−1 . Although g 7→ vgmvg−1 is not always norm-continuous, it is when m ∈ KB(E). More generally, there is also a natural action ofG on LB(E, F) given by t ·L =wt Lvt−1 . Again this action is only norm-continuouson KB(E, F), in general.

Given a G-algebra B and a Hilbert B,G-module (E, v) we make L2(G, E) intoa Hilbert B,G-module (L2(G, E), v⊗ λ), where

(v⊗ λ)t f (s)= vt f (t−1s).(2–2)

Let E be a Hilbert B,G-module. In the following we will denote by E∞ theHilbert B,G-module which is the direct sum of a sequence of copies of E , i.e. E∞

consists of the sequences (e1, e2, e3, . . . ) of elements in E for which∑

∞

i=1〈ei , ei 〉

converges in norm in B, and the G-action is the obvious one:

t · (e1, e2, e3, . . . )= (t · e1, t · e2, t · e3, . . . ).

We say that a Hilbert B,G-module E is countably generated when it is countablygenerated as a Hilbert B-module, i.e. when there is a countable set M ⊆ E suchthat the span of M B is dense in E . Since B is required to be σ -unital it is countablygenerated as a Hilbert B,G-module.

In the following we let K denote the C∗-algebra of compact operators on theHilbert space l2. A G-algebra B is stable when B ⊗ K ' B as G-algebras.

We denote by KG the C∗-algebra of compact operators on L2(G), which weconsider as a G-algebra; we have KG = (KG,Ad λ), where λ is the left-regularrepresentation of G.

Theorem 2.3 [Kasparov 1980b; Mingo and Phillips 1984]. Let B be a G-algebraand E a countably generated Hilbert B,G-module. Then

L2(G, E)⊕ L2(G, B∞)' L2(G, B∞)

as Hilbert B,G-modules.

Corollary 2.4. Let B be a G-algebra and E a countably generated Hilbert B,G-module. Assume B is stable. Then L2(G, E)⊕ L2(G, B) ' L2(G, B) as HilbertB,G-modules.

We denote by KG the G-algebra (K ⊗ KG, idK ⊗ Ad λ).

Definition 2.5. A G-algebra A will be called G-stable when A⊗KG is isomorphicto A as G-algebras.

Note that A ⊗ KG is always G-stable.

368 KLAUS THOMSEN

Lemma 2.6. Let A be a G-algebra. The following are equivalent:

(1) A is G-stable.

(2) A is stable and A ' A ⊗ KG as G-algebras.

(3) A is stable and A ' L2(G, A) as Hilbert A,G-modules.

Lemma 2.7. Let (B, β) be a G-stable G-algebra, and let u be a unitary represen-tation of G on l2. Then (B⊗K, β⊗Ad u) is ∗-isomorphic to (B, β) as G-algebras.

Theorem 2.8 [Meyer 2000]. Let A and B be G-algebras. Assume that A isG-stable. Let E be a countably generated Hilbert B,G-module and ϕ : A →

LB(E) an equivariant ∗-homomorphism such that ϕ(A)E = E . It follows thatE ⊕ L2 (G, B∞)' L2(G, B∞) as Hilbert B,G-modules.

Corollary 2.9. Let A and B be G-stable G-algebras. Let E be a countably gen-erated Hilbert B,G-module and ϕ : A → LB(E) an equivariant ∗-homomorphismsuch that ϕ(A)E = E . It follows that E ⊕ B ' B as Hilbert B,G-modules.

3. Stabilizing equivariant KK -theory: The even case

In this section we take the main steps towards a simplification in the definition ofequivariant KK -theory for G-algebras that are also G-stable. We concentrate onthe even case, i.e. on KK 0

G , since this is actually the most difficult case. The oddcase is easier and will be dealt with in the next section. What we do correspondsin the nonequivariant case to the substitution of general Hilbert C∗-modules by asingle canonical one; see for example [Blackadar 1986, Proposition 17.4.1]. Tosome extend, all we do is to show how R. Meyer’s construction [2000, Lemma3.3] can be made to work modulo operator homotopy and addition by degenerateelements, rather than homotopy.

Throughout this section A and B are G-algebras, A separable, B stable. Agraded Hilbert B,G-module is a graded Hilbert B-module E which is also a HilbertB,G-module with the same “inner product” such that the G-action of G on E com-mutes with the grading. An even Kasparov triple (E, ϕ, F) for A and B consists ofa graded Hilbert B,G-module E , an equivariant ∗-homomorphism ϕ : A → LB(E)mapping into the degree 0 elements of LB(E) and a degree 1 element F ∈ LB(E)such that

(F∗− F)ϕ(a), (F2

− 1)ϕ(a), Fϕ(a)−ϕ(a)F, (g · F − F)ϕ(a) ∈ KB(E)

for all a ∈ A and all g ∈ G. The even Kasparov triple (E, ϕ, F) is degenerate when

(F∗− F)ϕ(a)= (F2

− 1)ϕ(a)= Fϕ(a)−ϕ(a)F = (g · F − F)ϕ(a)= 0

for all g, a. Two even Kasparov triples (E0, ϕ0, F0) and (E1, ϕ1, F1) are operatorhomotopic when there is a family (E, ϕ,G t), t ∈ [0, 1], of even Kasparov triples


for A and B such that t 7→ G t is norm-continuous, (E, ϕ,G0) is isomorphic to(E0, ϕ0, F0) and (E, ϕ,G1) is isomorphic to (E1, ϕ1, F1).

By definition [Kasparov 1988] KK 0G(A, B) consists of the homotopy classes

of even Kasparov triples. It was pointed out in [Baaj and Skandalis 1989] thatKK 0

G(A, B), in line with more general equivariant KK -theory groups, can alsobe defined as the equivalence classes of even Kasparov triples for A and B, whenthe equivalence is operator homotopy after addition by degenerate elements, ratherthan homotopy as in [Kasparov 1988]. As in the nonequivariant case the equalitybetween the two definitions follows from the fact that the Kasparov product canbe defined modulo the apparently strongest of the two equivalence relations. Toobtain the description of KK 0

G(A, B) as the K1-group of a C∗-algebra we need towork entirely with the latter notion of equivalence for even Kasparov triples.

The Hilbert B,G-module B ⊕ B graded by the map (x, y) 7→ (x,−y) will bedenoted by Be. An even Kasparov triple (E, ϕ, F) for A and B will be calledelementary when E = Be and essential when ϕ(A)E = E . Note that the directsum of two elementary and/or essential even Kasparov triples are isomorphic to anelementary and/or essential even Kasparov triple.

Definition 3.1. An even Kasparov triple (E, ϕ, F) will be called homogeneouswhen it has the form E = E0 ⊕ E0, graded by (x, y) 7→ (x,−y), for some HilbertB,G-module E0, and ϕ = (ψ,ψ), where ψ : A → LB(E0) is an equivariant ∗-homomorphism.

Given a homogeneous even Kasparov triple (E, ϕ, F) = (E0 ⊕ E0, ϕ, F) asabove, there is a canonical way to form a homogeneous and degenerate even Kas-parov triple; namely, (E, ϕ, 1), where 1 ∈ LB(E) is

1 =(0

110

).

Lemma 3.2. Let E = (E, ϕ, F) be an even Kasparov triple. There is then a degen-erate even Kasparov triple D such that E⊕D is isomorphic to a homogeneous evenKasparov triple. When E is elementary we can choose D to be both elementary anddegenerate.

Proof. Let D0 = (Be, 0, 1). It follows from Kasparov’s stabilization theorem[1980b] that E ⊕ D0 is isomorphic to an even Kasparov triple of the form

(E0 ⊕ E0, (ψ+, ψ−), F0) ,

where E0 ⊕ E0 = B ⊕ B is graded by (x, y) 7→ (x,−y), and each of the twoB-summands carry an action by G which gives it the structure of a Hilbert B,G-module (not necessarily the canonical such structure). By adding to D0 the degen-erate even Kasparov triple (B ⊕ B, 0, 0) with the two G-actions interchanged wemay assume that the G-actions on the two B-summands agree (but not that they

370 KLAUS THOMSEN

are the canonical one). The infinite direct sum⊕

∞

1 E0 is a Hilbert B,G-module,and the triples

E+ =((⊕

∞

1 E0)⊕(⊕

∞

1 E0),(⊕

∞

1 ψ+

)⊕(⊕

∞

1 ψ+

),( 0

110

))and

E− =((⊕

∞

1 E0)⊕(⊕

∞

1 E0),(⊕

∞

1 ψ−

)⊕(⊕

∞

1 ψ−

),( 0

110

))are both degenerate even Kasparov triples. The direct sum E ⊕ D0 ⊕ E+ ⊕ E− isisomorphic to an even Kasparov triple of the form

(3–1)((⊕

∞

−∞E0)⊕(⊕

∞

−∞E0),8+ ⊕8−, F ′

),

where

8+ = (. . . . . . , ψ+, ψ+, ψ+, ψ−, ψ−, . . . . . . )

8− = (. . . . . . , ψ+, ψ+, ψ−, ψ−, ψ−, . . . . . . ).

Let S ∈ LB(⊕

∞

−∞E0)

be the two-sided shift; specifically, when e = (ei )i∈Z ∈⊕∞

−∞E0, S(e) is given by S(e)i = ei+1. Then

T =( 1

00S

)∈ LB

((⊕∞

−∞E0)⊕(⊕

∞

−∞E0))

is a G-invariant unitary of degree 0 such that Ad T ◦(8+⊕8−)=8+⊕8+. ThusT is an isomorphism between the Kasparov triple (3–1) and((⊕

∞

−∞E0)⊕(⊕

∞

−∞E0),8+ ⊕8+, T F ′T ∗

),

which is a homogeneous even Kasparov triple. It follows that D = D0 ⊕ E+ ⊕ E−

has the required property. When E is an elementary even Kasparov triple we cantake D0 =0, and then D will be (isomorphic to) an elementary even Kasparov triplesince B is stable. �

Let E be a Hilbert B,G-module. Let ϕ : A → LB(E) be an equivariant ∗-homomorphism. Since A is a Hilbert A-module in itself, we can form the internaltensor product A ⊗ϕ E which is a graded Hilbert B-module E ′; see, for instance,[Jensen and Thomsen 1991, 2.1.4]. Since ϕ is equivariant, E ′ is actually a gradedHilbert B,G-module in a canonical way; compare the proof of Theorem 2.8. Fol-lowing [Connes and Skandalis 1984], we introduce the notion of connections inthis setting. For every a ∈ A we can define an adjointable map Ta ∈ LB(E, E ′)

such that Ta(y) = a ⊗ϕ y. The adjoint T ∗a is determined by the condition that

T ∗a (b ⊗ϕ e)= ϕ(a∗b)e, and we set

Ta =

(0 T ∗

aTa 0

)∈ LB

(E ⊕ E ′

).


Let F ∈ LB(E). An F-connection is an element F ′∈ LB

(E ′)

such that

[Ta, F ⊕ F ′] ∈ KB

(E ⊕ E ′

)for all a ∈ A.

This condition is equivalent to

(3–2) Ta F − F ′Ta ∈ KB(E, E ′),

and

(3–3) FT ∗

a − T ∗

a F ′∈ KB(E ′, E)

for all a ∈ A. In particular, there is an F-connection, by [Connes and Skandalis1984]. When E is graded and F has degree 1 there is an F-connection of degree 1.

Lemma 3.3 [Meyer 2000, Lemma 3.1]. In the setting above, assume that A isG-stable, that ϕ(A)E = E , and that

[ϕ(a), F], (F∗− F)ϕ(a), (g · F − F) ϕ(a) ∈ KB(E)

for all a ∈ A and all g ∈ G. There is then a G-invariant F-connection. �

Lemma 3.4. In the setting above, assume that E = (E, ϕ, F) is an even Kasparovtriple. Define ϕ′

: A → LB(E ′)

by ϕ′(a)(a1 ⊗ϕ e)= aa1 ⊗ϕ e, and let F ′∈ LB

(E ′)

be an F-connection of degree 1. It follows that (E ′, ϕ′, F ′) is an even Kasparovtriple.

Proof. This is stated as part of [Meyer 2000, Lemma 3.3], but is really one ofthe fundamental steps in the construction of the Kasparov product of [E, ϕ, F] ∈

KK 0G(A, B) with [idA] ∈ KK 0

G(A, A). The details of the argument can be foundin [Jensen and Thomsen 1991, Lemma 2.2.6], for example. �

Note that the even Kasparov triple (E ′, ϕ′, F ′) of Lemma 3.4 is essential, andthat E ′ is isomorphic, as a graded Hilbert B,G-module, to Eess =ϕ(A)E under themap given by a ⊗ϕ e 7→ ϕ(a)e. Under this isomorphism F ′ turns into a degree 1operator which we denote by Fess. The defining relations for F ′, (3–2) and (3–3),turn into the conditions

(3–4) ϕ(a)F − Fessϕ(a) ∈ KB(E, Eess)

and

(3–5) Fϕ(a)−ϕ(a)Fess ∈ KB(Eess, E).

In particular we see that Fess is determined up to “a compact perturbation”, in thesense that ϕ(a)

(Fess − F ′′

)∈ KB(Eess) for all a ∈ A, when F ′′ is another operator

in LB(Eess) which satisfies (3–5).Set ϕess(a)= ϕ(a)|Eess , and note that (E ′, ϕ′, F ′) is then isomorphic, as an even

Kasparov triple, to the essential even Kasparov triple Eess = (Eess, ϕess, Fess). Note

372 KLAUS THOMSEN

also that Eess is both essential and homogeneous when E is homogeneous. It wasshown by Meyer [2000, Lemma 3.3] that E and Eess are homotopic and hencedefine the same element of KKG(A, B). We can therefore conclude that the evenKasparov triples E and Eess are operator homotopic after addition by degenerateeven Kasparov triples. Since we need to know what the involved degenerate tripleslook like, we have to obtain a more explicit proof of this fact. For this we need thefollowing lemma.

Lemma 3.5. Let A and B be a σ -unital G-algebras and ϕ : A → B an equivariantsurjection. Consider a separable closed self-adjoint subspace F ⊆ M(B) and afinite subgroup G0 ⊆ G. Then canonical extension ϕ : M(A) → M(B) maps{m ∈ M(A) :m f − f m ∈kerϕ, f ∈F, g·m−m ∈kerϕ, g ∈ G, g0·m =m, g0 ∈ G0}

onto M(B)G ∩ϕ(F)′.

Proof. This is an equivariant version of a result from [Olsen and Pedersen 1989].The proof presented in [Jensen and Thomsen 1991, Theorem 1.1.26] can be easilyadopted to the equivariant case by use of an asymptotically G-invariant approxi-mate unit. We leave the details to the reader. �

Lemma 3.6. Let E = (E, ϕ, F) be a homogeneous even Kasparov triple. SetZ1 = (Eess, 0, Fess)⊕ (E, ϕ, 1)⊕ (Eess, 0, 1)⊕ (E, 0, 1)⊕ (Eess, ϕess, 1) and Z2 =

(E, 0, F)⊕(E, ϕ, 1)⊕(Eess, 0, 1)⊕(E, ϕ, 1)⊕(Eess, 0, 1). Then E⊕Z1 is operatorhomotopic to Eess ⊕ Z2.

Proof. Consider the even Kasparov triple (E, ψ, H) for M2(A) and B, whereE = E ⊕ Eess, H = F ⊕ Fess, and

ψ : M2(A)→ LB(E ⊕ Eess)=

(LB(E) LB(Eess, E)

LB(E, Eess) LB(Eess)

)is given by

ψ

(a11 a12

a21 a22

)=

(ϕ(a11) ϕ(a12)

ϕ(a21) ϕ(a22)

).

For t ∈ [0, 1], set

Rt =

(−t

√1 − t2

√1 − t2 t

),

which we consider as a unitary multiplier of M2(A) in the obvious way. Set ψt =

ψ◦Ad Rt and ι(a)=(a

0

). Then (E, ψt◦ι, H), t ∈[0, 1], is a path of even Kasparov

triples connecting (E, ϕ, F)⊕(Eess, 0, Fess) and (E, 0, F)⊕(Eess, ϕ, Fess). Since

(E,ψ1 ◦ ι,H)⊕ (E,ψ ◦ ι, 1)⊕ (E,ψ0 ◦ ι, 1)

= (E,ϕ,F)⊕ (Eess,0,Fess)⊕ (E,ϕ, 1)⊕ (Eess,0, 1)⊕ (E,0, 1)⊕ (Eess,ϕess, 1)


and

(E,ψ0 ◦ ι,H)⊕ (E,ψ ◦ ι, 1)⊕ (E,ψ1 ◦ ι, 1)

= (E,0,F)⊕ (Eess,ϕess,Fess)⊕ (E,ϕ, 1)⊕ (Eess,0, 1)⊕ (E,ϕ, 1)⊕ (Eess,0, 1),

it suffices to show that (E, ψ1, H)⊕ (E, ψ, 1)⊕ (E, ψ0, 1) is operator homotopicto (E, ψ0, H)⊕ (E, ψ, 1)⊕ (E, ψ1, 1). Let p : LB(E) → LB(E)/KB(E) be thequotient map. Let X be the C∗-subalgebra of LB(E) generated by

KB(E)∪ψ(M2(A))⋃n∈N

H nψ(M2(A)).

Then H∗X ∪ X H∗∪ H X ∪ X H ⊆ X . We can therefore consider H as a multiplier

of X . Note that p(X) is generated by p ◦ψ(M2(A))⋃

n∈N p(H n)p ◦ψ(M2(A))and that p ◦ψ(M2(A))p(X) = p(X). In particular, it follows that p ◦ψ extendsto a unital ∗-homomorphism of the multiplier algebras:

p ◦ψ : M(M2(A))→ M(p(X)).

We set T t = p ◦ψ(Rt) ∈ M(p(X)), which is a symmetry for each t ∈ [0, 1].Observe that each T t is invariant under the action of Z2 ⊕ G, coming from thegrading of E and the representation of G, and that

T t p(H)p ◦ψ(a)= T t p ◦ψ(a)p(H)= p ◦ψ(Rt a)p(H)

= p(H)p ◦ψ(Rt a)= p(H)T t p ◦ψ(a)

for all t ∈ [0, 1], a ∈ M2(A). It follows that T t and p(H) commute in M(p(X)).Since T t , t ∈ [0, 1], is a norm-continuous path of unitaries in the connected com-ponent of 1 in the unitary group of M(p(X))G ∩ p(H)′, it follows from Lemma3.5 that we can find a norm-continuous path Tt , t ∈ [0, 1], of degree 0 unitaries inM(X) such that p(Tt)= T t , while g ·Tt −Tt , Tt H − H Tt ∈ KB(E) for all t ∈ [0, 1]

and all g ∈ G. Since KB(E) is an essential ideal in X there is a unique degree0 unitary St ∈ M(KB(E)) = LB(E) such that St x = Tt x for all x ∈ X . Since Tt

depends norm-continuously on t so does St , and

(3–6) St H − H St ∈ KB(E),

since this is true for Tt . In addition

(3–7) ψ(a)St −ψ(a Rt), Stψ(a)−ψ(Rt a) ∈ KB(E)

and

(3–8) g · St − St ∈ KB(E)

374 KLAUS THOMSEN

for all t ∈ [0, 1], g ∈ G and a ∈ M2(A). It follows that

(3–9) (E, ψ, S∗

t H St), t ∈ [0, 1],

is an operator homotopy. Let ∼OH denote an operator homotopy. By (3–8) and(3–7) there is an operator homotopy

(3–10) (E, ψ, S∗

t H St)⊕ (E, ψt , 1) ∼OH (E, ψ, 1)⊕ (E, ψt , H)

for all t ∈ [0, 1], obtained by rotating( H

1

)to( 1

H

). Then

(E,ψ0,H)⊕ (E,ψ,1)⊕ (E,ψ1,1)

∼OH (E,ψ,S∗

0 H S0)⊕ (E,ψ0,1)⊕ (E,ψ1,1) (by (3–10) applied with t = 0)

∼OH (E,ψ,S∗

1 H S1)⊕ (E,ψ0,1)⊕ (E,ψ1,1) (by (3–9))

∼OH (E,ψ0,1)⊕ (E,ψ,1)⊕ (E,ψ1,H) (by (3–10) applied with t = 1).

�

Lemma 3.7. Let A and B be G-algebras. Assume that B is G-stable and separa-ble. There is an equivariant ∗-homomorphism ϕ : A→ M(B) such that ϕ(A)B = B.

Proof. Since B is stable, B ' B ⊗ K as G-algebras. Let (π, u) be a covariantnondegenerate unitary representation of A on l2, i.e. π : A → B(l2) = M(K) is a∗-homomorphism, u is a continuous unitary representation of G on l2, π(A)l2 = l2

and ugπ(a)u∗g =π(g ·a) for all g, a. Such a pair (π, u) exists; see [Pedersen 1979],

for example. We can then define an equivariant ∗-homomorphism π0 : (A, α) →

(M(B⊗K), β⊗Ad u), where β is the canonical extension of the given action of Gon B, such that π0(a)(b ⊗ k)= b ⊗π(a)k. Note that π0(A)(B ⊗ K)= B ⊗ K. ByLemma 2.7 there is an equivariant ∗-isomorphism θ : (B ⊗K, β⊗Ad u)→ (B, β).Set ϕ = θ ◦π0, where θ : M(B ⊗ K)→ M(B) is the canonical extension of θ . �

Lemma 3.8. Let (E, ϕ, Ft) , t ∈ [0, 1], be an operator homotopy of even Kasparovtriples. It follows that there is an operator homotopy (Eess, ϕess, Ht) , t ∈ [0, 1],such that H0 = (F0)ess and H1 = (F1)ess.

Proof. This follows from the construction of an Ft -connection (see [Connes andSkandalis 1984] or [Jensen and Thomsen 1991, Proposition 2.2.5]), and the factthat Fess is unique up to compact perturbation. �

Theorem 3.9. Let A and B be G-algebras, A separable. Assume that A and B areG-stable.

(a) Every element of KK 0G(A, B) is represented by an even Kasparov triple for A

and B which is both elementary and essential.


(b) Two elementary and essential even Kasparov triples, E1 and E2, for A andB, define the same element of KK 0

G(A, B) if and only if there are degenerateeven Kasparov triples, D1 and D2, for A and B which are both elementaryand essential, such that E1 ⊕ D1 is operator homotopic to E2 ⊕ D2.

Proof. (a) By Lemma 3.2 and Lemma 3.6 every element of KKG(A, B) is repre-sented by an even Kasparov triple E = (E, ϕ, F) which is both homogeneous andessential. By Lemma 3.7 there is a ∗-homomorphism π : A → M(B) such thatπ(A)B = B. Then

(3–11) Z =(Be, π ⊕π,

( 01

10

))is a degenerate even Kasparov triple for A and B. It follows from Corollary 2.9that E ⊕ Be is isomorphic to Be as graded Hilbert B,G-modules. Thus E ⊕ Z isisomorphic to an elementary and essential even Kasparov triple for A and B.

(b) Let Ei = 1, 2, be elementary and essential even Kasparov triples for A and Brepresenting the same element in KK 0

G(A, B). It follows that there are degenerateeven Kasparov triples Di , i = 1, 2, such that E1 ⊕ D1 is operator homotopic toE2 ⊕ D2. It follows then from Lemma 3.8 that (E1 ⊕ D1)ess is operator homotopicto (E2 ⊕ D2)ess. Since Ei is essential, (Ei ⊕ Di )ess is isomorphic to Ei ⊕(Di )ess , i =

1, 2. It follows from Corollary 2.9 that (Di )ess ⊕ Z is isomorphic to a degenerateeven Kasparov triple which is both elementary and essential. This completes theproof. �

Theorem 3.10. Let A and B be G-algebras, A separable. Assume that A and Bare G-stable.

(a) Every element of KK 0G(A, B) is represented by an elementary even Kasparov

triple.

(b) Two elementary even Kasparov triples, X and Y, for A and B, define the sameelement of KK 0

G(A, B) if and only if there are degenerate and elementaryeven Kasparov triples, D1 and D2, for A and B, such that X ⊕ D1 is operatorhomotopic to Y ⊕ D2.

Proof. Part (a) follows from Theorem 3.9(a).

(b) It follows from Theorem 3.9(b) that it suffices to consider an elementary evenKasparov triple E and show that there are degenerate and elementary even Kasparovtriples Di , i = 1, 2, such that E ⊕ D1 is operator homotopic to X ⊕ D2, whereX is an elementary and essential even Kasparov triple. After an application ofLemma 3.2 we may assume that E is both elementary and homogeneous. We canthen consider the degenerate even Kasparov triples Zi , i = 1, 2, constructed fromE as in Lemma 3.6, so that E ⊕ Z1 is operator homotopic to Eess ⊕ Z2 by that

376 KLAUS THOMSEN

lemma. Note that because E is elementary, Z2 is (up to isomorphism) the sum oftwo degenerate even Kasparov triples — one which thanks to Corollary 2.9 can bestabilized in an equivariant way, namely,

(Eess, 0, 1

)⊕(Eess, 0, 1

), and another

which is (isomorphic to) an elementary degenerate even Kasparov triple, namely,(E, 0, F) ⊕ (E, ϕ, 1) ⊕ (E, ϕ, 1). It follows therefore from Corollary 2.9 thatZ2 ⊕ Z is isomorphic to an elementary degenerate even Kasparov triple, whereZ is the even Kasparov triple (3–11). Similarly, Z1 ⊕ Z is also isomorphic to anelementary degenerate even Kasparov triple. Since E ⊕ Z2 ⊕ Z ⊕ Z is operatorhomotopic to Eess ⊕ Z1 ⊕ Z ⊕ Z, and since Eess ⊕ Z is isomorphic to an essentialand elementary even Kasparov triple by Lemma 3.6, we can take D1 to be anisomorphic copy of Z1 ⊕ Z ⊕ Z, X to be an isomorphic copy of Eess ⊕ Z, and D2

to be an isomorphic copy of Z2 ⊕ Z. �

4. Stabilizing equivariant KK -theory: The odd case

This section is the odd analog of the previous section. The development is basicallyidentical with the even case, but there are a few more shortcuts that we can exploit.The first comes from the description of KK 1

G given in [Thomsen 2000].Throughout this section A and B are G-algebras, A separable and B stable. An

odd Kasparov triple (E, ϕ, P) for A and B consists of a Hilbert B,G-module E ,an equivariant ∗-homomorphism ϕ : A → LB(E) and an element P ∈ LB(E) suchthat

(P∗− P)ϕ(a), (P2

− P)ϕ(a), Pϕ(a)−ϕ(a)P, (g · P − P)ϕ(a) ∈ KB(E)

for all a ∈ A and all g ∈ G. The odd Kasparov triple (E, ϕ, P) is degenerate when

(P∗− P)ϕ(a)= (P2

− P)ϕ(a)= Pϕ(a)−ϕ(a)P = (g · P − P)ϕ(a)= 0

for all g, a. Two odd Kasparov triples (E0, ϕ0, P0) and (E1, ϕ1, P1) are operatorhomotopic when there is a family (E, ϕ,G t), t ∈ [0, 1], of odd Kasparov triplesfor A and B such that t 7→ G t is norm-continuous, (E, ϕ,G0) is isomorphic to(E0, ϕ0, P0) and (E, ϕ,G1) is isomorphic to (E1, ϕ1, P1).

The odd equivariant KK -group KK 1G(A, B) consists of the homotopy classes

of odd Kasparov triples. As in the even case, KK 1G(A, B) can also be defined as

the equivalence classes of odd Kasparov triples for A and B, when the equivalenceis operator homotopy after addition by degenerate elements. This description ofKK 1

G(A, B) differs from Kasparov’s original definition [1988], but it is not difficultto see that they are equivalent. In fact, as shown in [Thomsen 2000], it suffices toconsider Hilbert B,G-modules E , which as a Hilbert B-module is B itself, butwith a “twisted” G-action.


We say that an odd Kasparov triple (E, ϕ, F) is essential when ϕ(A)E = E andelementary when E = B.

Let E = (E, ϕ, P) be an odd Kasparov triple. Set E ′= A ⊗ϕ E , and define

ϕ′: A → LB

(E ′)

by ϕ′(a)(a1 ⊗ϕ e) = aa1 ⊗ϕ e, and let P ′∈ LB

(E ′)

be a P-connection. It follows that (E ′, ϕ′, P ′) is an odd Kasparov triple. So see this, onecan for example apply Lemma 3.4 to the even Kasparov triple(

E ⊕ E, (ϕ, ϕ), (2P − 1, 1 − 2P)),

where E ⊕ E is graded by (x, y) 7→ (y, x), or take a look at [Jensen and Thomsen1991, Lemma 2.2.6]. Set Eess =ϕ(A)E, ϕess(a)=ϕ(a)|Eess , and let Pess ∈LB (Eess)

be the image of P ′ under the isomorphism A⊗ϕ E ' Eess given by a⊗ϕ e 7→ϕ(a)e.Then (Eess, ϕess, Pess) is an essential odd Kasparov triple. The following lemma isproved in the same way as Lemma 3.6.

Lemma 4.1. Let E = (E, ϕ, P) be an odd Kasparov triple. Set

Z1 = (Eess, 0, Pess)⊕ (E, ϕ, 0)⊕ (Eess, 0, 0)⊕ (E, 0, 0)⊕ (Eess, ϕess, 0),

Z2 = (E, 0, P)⊕ (E, ϕ, 0)⊕ (Eess, 0, 0)⊕ (E, ϕ, 0)⊕ (Eess, 0, 0).

Then E ⊕ Z1 is operator homotopic to Eess ⊕ Z2.

Similarly, the proofs of Theorem 3.9 and Theorem 3.10 carry over to the oddcase with only the obvious changes. E.g. the even Kasparov triple Z which playsa prominent role in both proofs can be substituted by the odd Kasparov triple(B, π, 0), where π : A → M(B) is an equivariant ∗-homomorphism such thatB = π(A)B; see Lemma 3.7.

Theorem 4.2. Let A and B be G-stable G-algebras, with A separable.

(a) Every element of KK 1G(A, B) is represented by an odd Kasparov triple for A

and B which is both elementary and essential.

(b) Two elementary and essential odd Kasparov triples, E1 and E2, for A and B,define the same element of KK 1

G(A, B) if and only if there are degenerate oddKasparov triples, D1 and D2, for A and B which are both elementary andessential, such that E1 ⊕ D1 is operator homotopic to E2 ⊕ D2.

Theorem 4.3. Let A and B be G-stable G-algebras, with A separable.

(a) Every element of KK 1G(A, B) is represented by an elementary odd Kasparov

triple.

(b) Two elementary odd Kasparov triples, X and Y, for A and B, define the sameelement of KK 1

G(A, B) if and only if there are degenerate and elementaryodd Kasparov triples, D1 and D2, for A and B, such that X ⊕ D1 is operatorhomotopic to Y ⊕ D2.

378 KLAUS THOMSEN

5. Equivariantly absorbing homomorphisms

In this section we define and establish the existence of absorbing ∗-homomorphismsin the equivariant case. The main tools are Kasparov’s methods [1980b; 1988], theideas from [Thomsen 2001], and Corollary 2.9 above.

When X is a normed vector space with an action of G by linear transformationswe will in the following say that a net {xα}⊆ X is asymptotically G-invariant whenlimα g · xα− xα = 0, uniformly on compact subsets of G. Note that it follows from[Kasparov 1988, page 152, lemma] that a G-algebra always contains an asymp-totically G-invariant approximate unit with other convenient properties. We stateonly the part that we shall need.

Lemma 5.1 [Kasparov 1988]. Let A be a G-algebra, and D ⊆ M(A) a separableC∗-subalgebra. Then A contains an approximate unit {un} which is asymptoticallyG-invariant and asymptotically commutes with D, i.e. limn→∞ und − dun = 0 forall d ∈ D.

The multiplier algebra M(B) of a G-algebra is not always a G-algebra becausethe action of G on M(B) is only continuous for the strict topology in general. Theset of elements on which the G-action is continuous is a C∗-subalgebra of M(B),and it is a G-algebra. We denote this G-algebra by M(B)G . The image of M(B)Gin Q(B) will be denoted by Q(B)G . In contrast the fixed point algebra in M(B)and Q(B) will be denoted by M(B)G and Q(B)G , respectively.

Let E be a Hilbert B-module. A ∗-homomorphism π : M(A) → LB(E) willbe called strictly continuous, when π is strictly continuous on norm-bounded sets.A ∗-homomorphism π : A → LB(E) will be called quasi-unital when there is aprojection p ∈ LB(E) such that pE = π(A)E . It is known that π : A → LB(E) isquasi-unital if and only if π admits a strictly continuous extension π : M(A) →

LB(E); see [Lance 1995].

Theorem 5.2. Let A and B be G-algebras, A separable and B G-stable. Lets : A → M(B)G be a completely positive contraction. There is then a strictlycontinuous equivariant ∗-homomorphism π : M(A) → M(B) and a sequence ofcontractions {Wn} in M(B) such that limn→∞ W ∗

n π(a)Wn = s(a) for all a ∈ A.

Proof. Let α and β be the given actions of G on A and B, respectively. We beginby constructing two Hilbert B,G-modules, E0 and E . Give the algebraic tensorproduct A ⊗ Cc(G, B) a right B-module structure such that (a ⊗ f )b = a ⊗ f bwhere f b(g)= f (g)βg(b) and make it into a semi-inner-product B-module in thesense of [Lance 1995], such that

〈a ⊗ψ, a1 ⊗ψ1〉 =

∫G

g−1·(ψ(g)∗s(g · (a∗a1))ψ1(g)

)dg.


Let E0 denote the resulting Hilbert B-module, obtained by completion. Let r bethe right-regular representation of G, namely (rkψ) (g) = 1(k)1/2ψ(gk). Definea representation T of G on E0 such that Tk = αk ⊗ rk on A ⊗ Cc(G, B). ThenE0 is a Hilbert B,G-module which is countably generated since A is separable,G second countable and B σ -unital. Give next Cc(G, A)⊗ Cc(G, B) the right B-module structure such that ( j ⊗ f )b = j ⊗ f b, and turn it into a semi-inner-productB-module such that

〈 j ⊗ f, j1 ⊗ f1〉 =

∫G

∫G

g−1·(

f (g)∗s(g · ( j (g−1h)∗ j1(g−1h))) f1(g))

dg dh.

By completion we obtain a Hilbert B-module E . Set (α⊗λ)k j (g)= αk( j (k−1g)),and define a representation S of G as linear operators on E such that

Sk( j ⊗ f )= (α⊗ λ)k ( j)⊗ rk( f ).

Define a linear map8 :Cc(G, A)⊗Cc(G, B)→Cc(G, E0) such that8( j⊗ f )(h)=j (h)⊗ f . Then∫

G

⟨8( j ⊗ f )(h),8( j1 ⊗ f1)(h)

⟩dh = 〈 j ⊗ f, j1 ⊗ f1〉;

hence 8 extends to an isomorphism 8 : E → L2(G, E0) of Hilbert B,G-modules.Since 8 ◦ Sk = (λk ⊗ Tk) ◦8, we see that 8 itself is an isomorphism of HilbertB,G-modules. Define π ′

: M(A)→ LB(E) such that

π ′(a)( j ⊗ f )= π0(a) j ⊗ f,

where (π0(a) j)(g)= aj (g). Then π ′ is a strictly continuous equivariant ∗-homo-morphism. Let κn, n ∈ N, be a sequence of nonnegative functions in Cc(G) suchthat supp κn+1 ⊆ supp κn and

∫G κn(g)2 dg = 1 for all n, and⋂

n

supp κn = {e}.(5–1)

Let {un} be an approximate unit in A. For n,m ∈ N, set Wn,mb = κa⊗ κb

n ∈

Cc(G, A) ⊗ Cc(G, B), where κa(g) = κ1(g)un , and κbm(g) = κm(g)βg(b). We

claim that Wn,m is adjointable, as a map Wn,m : B → E . To see this, define firstT : Cc(G, A)⊗ Cc(G, B)→ B such that

T ( j ⊗ f )=

∫G

∫Gκ1(g−1h)κm(g)g−1

·(s(g · (un j (g−1h))) f (g)

)dg dh.

It is then straightforward to check that

〈Wn,mb, j ⊗ f 〉 = 〈b, T ( j ⊗ f )〉.(5–2)

380 KLAUS THOMSEN

Let∑

i ji ⊗ fi be a finite sum of simple tensors in Cc(G, A) ⊗ Cc(G, B). Byusing the Schwarz inequality for completely positive maps, as in [Lance 1995], forinstance, we find that∥∥∥∥T(∑

i

ji ⊗ fi

)∥∥∥∥=

∥∥∥∥∫G

∫Gκ1(g−1h)κm(g)

∑i

g−1·(s(g · (un ji (g−1h))) fi (g)

)dg dh

∥∥∥∥≤

∥∥∥∥∫G

∫G

g−1·

(∑i

fi (g)∗s(g · ( ji (g−1h)∗un))

×

∑l

s(g · (un jl(g−1h))) fl(g))

dg dh∥∥∥∥1/2

×

(∫G

∫Gκ1(g−1h)2κm(g)2 dg dh

)1/2

≤

∥∥∥∥∫G

∫G

g−1·

(∑i,l

fi (g)∗s(g · ( ji (g−1h)∗u2n jl(g−1h))) fl(g)

)dg dh

∥∥∥∥1/2

≤

∥∥∥∥∫G

∫G

g−1·

(∑i,l

fi (g)∗s(g · ( ji (g−1h)∗ jl(g−1h))) fl(g))

dg dh∥∥∥∥1/2

=

∥∥∥∥∑i

ji ⊗ fi

∥∥∥∥.It follows that T extends to a linear contraction T : E → B which then, by (5–2),is W ∗

n,m . Note that

(5–3) W ∗

n,mπ′(a)Wn,mb =

∫G

∫Gκ1(g−1h)2κm(g)2g−1

· (s(g · (una))) b dg dh,

so that

‖W ∗

n,mπ(a)Wn,mb − s(a)b‖

=

∥∥∥∥∫G

∫Gκ1(g−1h)2κm(g)2

(g−1(s(g · (una)))− s(a)

)b dg dh

∥∥∥∥≤ ‖b‖ sup{‖g−1

· s(g · (una))− s(a)‖ : g ∈ supp κm}

≤ ‖b‖‖s(una)− s(a)‖ +‖b‖ sup{‖g−1· s(g · (una))− s(una)‖ : g ∈ supp κm}

for all b ∈ B, a ∈ A. Moreover g 7→ g−1· s(g · a) is norm-continuous because s

takes values in M(B)G . Thus it follows from (5–1) and the preceding bound that

limn→∞

∥∥W ∗

n,knπ(a)Wn,kn − s(a)

∥∥= 0


for all a ∈ A, when k1 < k2 < k3 < · · · is an appropriately chosen sequence inN. Since E ⊕ B ' L2(G, E0)⊕ L2(G, B) ' L2(G, B) ' B by Corollary 2.4 andLemma 2.6, there is an equivariant adjointable isometry S : E → B of HilbertB,G-modules. Set π(a)= Sπ ′(a)S∗ and Wn = SWn,kn . �

Proposition 5.3. Let A and B be G-stable G-algebras. Assume that A is separa-ble. Let ϕ : A→ M(B) be an equivariant completely positive contraction. It followsthat there is a strictly continuous equivariant ∗-homomorphism π : M(A)→ M(B)and an asymptotically G-invariant sequence {Sn} of contractions in M(B) suchthat

limn→∞

S∗

nπ(a)Sn = ϕ(a)

for all a ∈ A.

Proof. Make A ⊗ B into a pre-Hilbert B-module such that (a ⊗ b)b1 = a ⊗ bb1

and such 〈a ⊗ b, a1 ⊗ b1〉 = b∗ϕ(a∗a1)b1. Let E denote the resulting Hilbert B-module. E is countably generated since A is separable and B σ -unital. Becauseϕ is equivariant we can make E into a Hilbert B,G-module by introducing therepresentation T of G on E given by Tk(a ⊗ b) = k · a ⊗ k · b. Let {un} be anasymptotically G-invariant approximate unit in A, as in Lemma 5.1, and defineVn : B → E by Vnb = un ⊗ b. Then Vn is an adjointable contraction such that

V ∗

n (a ⊗ b)= ϕ(una)b.

Since Tk Vnk−1· b = k · un ⊗ b, we see that {Vn} is asymptotically G-invariant

because {un} is. Define π0 : M(A) → LB(E) such that π0(m)(a ⊗ b) = ma ⊗ b,and note that π0 is a strictly continuous equivariant ∗-homomorphism, and thatπ0(A)E = E . Furthermore, limn→∞ V ∗

n π0(a)Vn = limn→∞ ϕ(unaun) = ϕ(a) forall a ∈ A. Since A and B are G-stable it follows from Corollary 2.9 that E ⊕ B ' Bas Hilbert B,G-modules. It follows that there is an adjointable equivariant isometryW : E → B. Set Sn = W Vn and π(m)= Wπ0(m)W ∗. �

Let A and B be G-algebras, B stable. Then M(B) contains a pair of G-invariantisometries V1, V2 with the property that V1V ∗

1 + V2V ∗

2 = 1, since this is clearly thecase of B ⊗ K. Using these isometries we can add maps from A to M(B) in theusual way: (ϕ⊕ψ) (a)= V1ϕ(a)V ∗

1 +V2ψ(a)V ∗

2 . The map ϕ⊕ψ is equivariant iffϕ and ψ both are. Note that this addition is independent of the choice of isometriesV1 and V2 up to conjugation by a G-invariant unitary. Since B is stable there is asequence {Si }

∞

i=1 of G-invariant isometries in M(B) such that∑

∞

i=1 Si S∗

i = 1 inthe strict topology. An equivariant ∗-homomorphism π : A → M(B) is saturatedwhen there is a G-invariant unitary U in M(B) such that

(5–4) Uπ(a)U∗=

∞∑i=1

S2iπ(a)S∗

2i

382 KLAUS THOMSEN

for all a ∈ A. Thus a saturated ∗-homomorphism A → M(B) is one which isunitarily equivalent to the sum of infinitely many copies of itself plus the zerohomomorphism. From the technical point of view, one of the main features ofa saturated ∗-homomorphism, π , that we shall need is that there is a sequenceWi , i ∈ N, of G-invariant isometries in M(B) such that W ∗

i W j = 0 when i 6= j ,W ∗

i π(a)W j = δ(i, j)π(a) for all a, i, j , and such that limi→∞ W ∗

i b = 0 for allb ∈ B. Indeed, in the notation of (5–4), Wi = U∗S2i will be such a sequence.

Any equivariant ∗-homomorphism can be saturated: When ϕ : A → M(B) is anequivariant ∗-homomorphism, ϕ′( · ) =

∑∞

i=1 S2iϕ( · )S∗

2i is saturated, and we callit the saturation of ϕ.

Theorem 5.4. Let A and B be G-algebras, A separable, B stable. Let π : A →

M(B) be a saturated equivariant ∗-homomorphism. Consider the following fiveconditions on π :

(1) For any completely positive equivariant contraction ϕ : A → M(B) there isan asymptotically G-invariant sequence of contractions {Wn} ⊆ M(B) suchthat limn→∞ ‖ϕ(a)− W ∗

n π(a)Wn‖ = 0 for all a ∈ A.

(2) For any completely positive equivariant contraction ϕ : A → M(B) there isan asymptotically G-invariant sequence {Vn} of isometries in M(B) such that

(a) V ∗n π(a)Vn −ϕ(a) ∈ B, n ∈ N , a ∈ A ,

(b) limn→∞ ‖V ∗n π(a)Vn −ϕ(a)‖ = 0, a ∈ A,

(c) g · Vn − Vn ∈ B for all n ∈ N and all g ∈ G.

(3) For any completely positive equivariant contraction ϕ : A → M(B) there isan asymptotically G-invariant and norm-continuous path, Vt , t ∈ [1,∞), ofisometries in M(B) such that

(a) V ∗t π(a)Vt −ϕ(a) ∈ B, t ∈ [1,∞[ , a ∈ A ,

(b) limt→∞ ‖V ∗t π(a)Vt −ϕ(a)‖ = 0, a ∈ A.

(c) g · Vt − Vt ∈ B for all t ∈ [1,∞[ and all g ∈ G.

(4) For any equivariant ∗-homomorphism ϕ : A → M(B) there is an asymptot-ically G-invariant and norm-continuous path, Ut , t ∈ [1,∞), of unitaries inM(B) such that

(a) Ut(π(a)⊕ϕ(a))U∗t −π(a) ∈ B, t ∈ [1,∞), a ∈ A,

(b) limt→∞ ‖Ut(π(a)⊕ϕ(a))U∗t −π(a)‖ = 0, a ∈ A ,

(c) g · Ut − Ut ∈ B for all t ∈ [1,∞) and all g ∈ G.

(5) For any equivariant ∗-homomorphism ϕ : A → M(B) there is an asymptoti-cally G-invariant sequence {Un} of unitaries Un ∈ M(B) such that

limn→∞

‖Un(π(a)⊕ϕ(a))U∗

n −π(a)‖ = 0, a ∈ A.


Then (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5). When, in addition, A and B are G-stable, (5)⇒ (1) so that (1)-(5) are equivalent in this case.

Proof. (1) ⇒ (2): Let ϕ : A → M(B) be a completely positive equivariant contrac-tion. By (1) there is an asymptotically G-invariant sequence {Wn} of contractionsin M(B) such that limn→∞ W ∗

n π(a)Wn = ϕ(a), a ∈ A. Set

Xn =

(Wn 0√

1 − W ∗n Wn 0

)which is a partial isometry in M2(M(B)) such that X∗

n Xn =( 1

000

)for all n and

limn→∞

X∗

n(π(a)

0

)Xn =

(ϕ(a)

0

), a ∈ A.

Furthermore, {Xn} is asymptotically G-invariant since {Wn} is. Since B is stablethere is a G-invariant element V ∈ M2(M(B)) such that V V ∗

=1 and V ∗V =( 1

000

).

Then Rn = XnV ∗ is an isometry and

limn→∞

R∗

n(π(a)

0

)Rn = V

(ϕ(a)

0

)V ∗, a ∈ A.

Set

U =

(V ∗ 1−V ∗V0 V

),

which is a G-invariant unitary in M4(M(B)) with the property that

U(

V(ϕ(a)

0

)V ∗

0

)U∗

=

(ϕ(a)

00

0

).

Set Sn =( Rn

Rn

)U∗ which is an isometry in M4(M(B)) for all n such that

limn→∞

S∗

n

(π(a)

00

0

)Sn =

(ϕ(a)

00

0

)for all a ∈ A. {Sn} is asymptotically G-invariant since {Wn} is. Since B3

' B asHilbert B,G-modules we can use an isomorphism M4(B)' M2(B) of G-algebrasto get an asymptotically G-invariant sequence {S′

n} of isometries in M(M2(B))such that

limn→∞

S′

n∗(π(a)

0

)S′

n =(ϕ(a)

0

)for all a ∈ A. Since B is stable there are G-invariant isometries V1, V2 ∈ M(B)such that V1V ∗

1 + V2V ∗

2 = 1. Let 2 : M2(M(B)) → M(B) be the G-equivariant∗-isomorphism given by

(5–5) 2

(a11 a12

a21 a22

)=

∑i, j

Vi ai j V ∗

j .

384 KLAUS THOMSEN

Set T ′n = 2(S′

n). Then 2 ◦(ϕ

0

)= V1ϕ( · )V ∗

1 and {T ′n} is an asymptotically G-

invariant sequence of isometries such that limn→∞ T ′n∗V1π(a)V ∗

1 T ′n = V1ϕ(a)V ∗

1for a ∈ A. Since π is saturated there is a G-invariant unitary U such that Uπ(a)U∗

=

V1π(a)V ∗

1 . Then Tn = U∗T ′nV1, n ∈ N, is an asymptotically G-invariant sequence

of isometries in M(B) such that limn→∞ T ∗n π(a)Tn = ϕ(a) for all a ∈ A. Note that

since π is saturated we can arrange that limn→∞ ‖T ∗n b‖ = 0 for all b ∈ B, and that

T ∗

i T j = 0, T ∗

i π(A)T j = {0}, for i 6= j .Fix a compact subset X with dense span in A, an ε > 0 and a compact subset

K ⊆ G. Let K = K1 ⊆ K2 ⊆ K3 ⊆ · · · be a sequence of compact sets such thatG =

⋃n Kn . Let {ei }

∞

i=1 be an approximate unit for B which is asymptoticallyG-invariant and asymptotically commutes with ϕ(A); see Lemma 5.1. Let n1 <

n2 < n3 < · · · be a sequence in N and set

f1 = e1/2n1, fk = (enk − enk−1)

1/2, k ≥ 2.

Let b be a strictly positive element in B. Assume that {ni } increases so fast that

‖g · fi − fi‖ ≤ 2−iε(5–6)

for all g ∈ Ki and all i ∈ N,

‖ fkb‖ ≤ 2−k, k ≥ 2,(5–7)

and

‖ fkϕ(a)−ϕ(a) fk‖ ≤ 2−kε(5–8)

for all a ∈ X and k ∈ N. Now let {Wk} be a subsequence of {Tk} chosen such that

‖g · Wi − Wi‖ ≤ 2−iε, g ∈ Ki ,(5–9)

‖W ∗

i b‖ ≤ 2−iε,(5–10)

and ∥∥W ∗

i π(a)Wi −ϕ(a)∥∥≤ 2−iε(5–11)

for all a ∈ X and all i ∈ N. It follows then from (5–10) and (5–7) that∑

∞

k=1 Wk fk

converges in the strict topology to an isometry W in M(B). Since

‖g · (Wk fk)− Wk fk‖ ≤ 2−k+1ε, k ≥ n, g ∈ Kn,

by (5–6) and (5–9) we see that g ·W −W ∈ B for all g ∈ G and that ‖g ·W −W‖ ≤

2ε, g ∈ K . Note that W ∗π(a)W =∑

∞

i=1 fi W ∗

i π(a)Wi fi for all a. Furthermore,(5–8) ensures that ϕ(a)−

∑∞

i=1 fiϕ(a) fi ∈ B and that ‖ϕ(a)−∑

∞

i=1 fiϕ(a) fi‖ ≤

ε for all a ∈ X . It follows then from (5–11) that W ∗π(a)W − ϕ(a) ∈ B and


‖W ∗π(a)W − ϕ(a)‖ ≤ ε for all a ∈ X . Since X, K and ε > 0 were arbitrary, (2)follows.

(2) ⇒ (3): Since π is saturated there is a sequence Wi , i ∈ N, of G-invariantisometries in M(B) such that W ∗

i W j = 0 when i 6= j and W ∗

i π(a)W j = δ(i, j)π(a)for all a, i, j . So by substituting WnVn for Vn , we may assume that V ∗

i V j = 0 andV ∗

i π(a)V j = 0 for all a ∈ A when i 6= j . Set Vt =√

t − iVi+1 +√

i + 1 − tVi ,for t ∈ [i, i + 1]. Then Vt , t ∈ [1,∞), is an asymptotically G-invariant and norm-continuous path of isometries such that (3a)–(3c) hold.

(3) ⇒ (4): In the following we will write ψ1 ∼ ψ2 between equivariant ∗-homo-morphisms ψ1, ψ2 : A → M(B), when there is an asymptotically G-invariant andnorm-continuous path Wt , t ∈ [1,∞), of unitaries in M(B) such that

limt→∞

Wtψ1(a)W ∗

t = ψ2(a)

for all a ∈ A and such that Wtψ1(a)W ∗t − ψ2(a) ∈ B and g · Wt − Wt ∈ B for

all a, t, g. Let ϕ : A → M(B) be an equivariant ∗-homomorphism. We needto show that π ⊕ ϕ ∼ π . Let {Vt } be an asymptotically G-invariant and norm-continuous path of isometries such that (3a), (3b) and (3c) hold. By consideringthe asymptotically G-invariant path of unitaries in M2(M(B)) given by

Ut =

(V ∗

t 1 − V ∗t Vt

0 Vt

),

we see that π ⊕ 0 ∼ ϕ⊕ 0. Note that π ∼ π ⊕ 0 ∼ π ⊕π ⊕ 0 since π is saturated.It follows that

π ⊕ϕ ∼ π ⊕ 0 ⊕ϕ ∼ π ⊕π ⊕ 0 ∼ π.

(4) ⇒ (5) is trivial.

(5) ⇒ (1) (when A and B are G-stable): It follows from Proposition 5.3 that there isan equivariant ∗-homomorphism µ : A → M(B) and an asymptotically G-invariantsequence of isometries {Wn} ⊆ M(B) such that limn→∞ W ∗

nµ(a)Wn = ϕ(a) for alla ∈ A. Let F1 ⊆ F2 ⊆ F3 ⊆ · · · be a sequence of finite subsets with dense unionin A and let K1 ⊆ K2 ⊆ K3 ⊆ · · · be a sequence of compact subsets in G whoseunion is G. Fix n ∈ N. It follows from (5) that there is a unitary Un ∈ M(B) suchthat

∥∥Un(π(a)⊕µ(a))U∗n −π(a)

∥∥≤ 1/n for all a ∈ Fn and ‖g · Un − Un‖ ≤ 1/nfor all g ∈ Kn . Let V1 and V2 be the isometries used to define the addition and setSn = UnV2Wn ∈ M(B). Then ‖Sn‖ ≤ 1, ‖g · Sn − Sn‖ ≤ 1/n + ‖g · Wn − Wn‖ forall g ∈ Kn , and ∥∥ϕ(a)− S∗

nπ(a)Sn∥∥≤

1n

+∥∥W ∗

nµ(a)Wn −ϕ(a)∥∥

for all a ∈ Fn . �

386 KLAUS THOMSEN

A saturated equivariant ∗-homomorphism π : A → M(B) will be called equiv-ariantly absorbing when it satisfies condition (5) of Theorem 5.4.

Proposition 5.5. Let A and B be G-algebras, A separable. Assume that A and Bare both G-stable. If ϕ : A → M(B) and π : A → M(B) are both equivariantlyabsorbing there is a norm-continuous path ut , t ∈ [0,∞), of unitaries in M(B)such that

(i) utπ(a)u∗t −ϕ(a) ∈ B for all a, t ,

(ii) g · ut − ut ∈ B for all g, t ,

(iii) limt→∞ utπ(a)u∗t −ϕ(a)= 0 for all a, and

(iv) limt→∞ g · ut − ut = 0, uniformly on compact subsets of G.

Proof. This follows from Theorem 5.2. �

Note that the condition for an equivariant ∗-homomorphism A → M(B) to beequivariantly absorbing does not reduce to the condition that it is absorbing, in thesense of [Thomsen 2001], when G = 0. The reason is that we require an equivari-antly absorbing ∗-homomorphism to be saturated, and in the nonequivariant casethis means that it must be unitarily equivalent to the infinite sum of copies of itselfplus the zero homomorphism. While this may not be the case of all absorbing ∗-homomorphisms, the additional requirement seems not to have any significance inpractice, and as demonstrated by Proposition 5.5, saturation is a very convenientproperty to have.

Proposition 5.6. Let A and B be C∗-algebras, A separable, B σ -unital and stable.Let π : A → M(B) be a saturated ∗-homomorphism with the following property:

(A) For any completely positive contraction ϕ : A → B there is a sequence ofcontractions {Wn} ⊆ M(B) such that limn→∞ ‖ϕ(a)− W ∗

n π(a)Wn‖ = 0 forall a ∈ A.

It follows that π is absorbing.

Proof. By combining the G =0 case of Theorem 5.4 with [Thomsen 2001, Theorem2.5] we see that it suffices to show that π has the following property:

(B) For any completely positive contraction ψ : A → M(B) there is a sequenceof contractions {Wn} ⊆ M(B) such that limn→∞ ‖ψ(a)− W ∗

n π(a)Wn‖ = 0for all a ∈ A.

To this end, let F ⊆ A be a finite set, and let ε > 0 be given. Choose a positiveelement u ∈ A such that ‖uau−a‖≤ε for all a ∈ F , and set F ′

={u2}∪{uau :a ∈ F}.

Let X be a compact subset of A which contains F ′ and spans a dense subspace inA. Let b be a strictly positive element of B. Let {ei }

∞

i=1 be an approximate unitfor B which asymptotically commutes with ψ(A). Let n1 < n2 < n3 < · · · be a


sequence in N and set f1 = e1/2n1 and fk = (enk −enk−1)

1/2 for k ≥ 2. We can arrangethat {ni } increases so fast that ‖ fkb‖≤ 2−k, k ≥ 2, and ‖ fkψ(a)−ψ(a) fk‖≤ 2−kε

for all a ∈ X and all k ∈ N. It follows then that∑

∞

i=1 fiψ(a) fi converges in thestrict topology for all a ∈ A, and that

(5–12)∥∥ ∞∑

i=1

fiψ(a) fi −ψ(a)∥∥≤ ε

for all a ∈ F ′. Since π has property (A) we can find a contraction Wi ∈ M(B) suchthat

(5–13)∥∥∥∥W ∗

i π(a)Wi − f 1/2i ψ(a) f 1/2

i

∥∥∥∥≤ 2−iε

for all a ∈ F ′. Since π is saturated we can arrange that W ∗

i W j = 0, W ∗

i π(A)W j =

{0} if i 6= j , and ‖W ∗

i b‖ ≤ 2−i for all i . Since ‖ f 1/2i b‖ ≤

√‖b‖ 2−i/2 for i ≥ 2,

we see that the sum W =∑

∞

i=1 Wi f 1/2i converges in the strict topology. Thanks to

the properties of the Wi we find that W ∗π(a)W =∑

∞

i=1 f 1/2i W ∗

i π(a)Wi f 1/2i for

all a ∈ A. It follows then from (5–13) and (5–12) that

(5–14) ‖W ∗π(a)W −ψ(a)‖ ≤ 2ε

for all a ∈ F ′. Set V = π(u)W , and note that it follows from (5–14) that ‖V ‖ ≤√

1 + 2ε. Furthermore, since ‖uau − a‖ ≤ ε for a ∈ F , we conclude from (5–14)that ‖V ∗π(a)V −ψ(a)‖ ≤ 3ε for all a ∈ F . It follows that π has property (B), asdesired. �

Theorem 5.7. Let A and B be separable G-algebras, both G-stable. There existsa strictly continuous equivariant ∗-homomorphism π : M(A) → M(B) such thatπ |A is both absorbing and equivariantly absorbing.

Proof. By [Thomsen 2001, Lemma 2.3] there is a sequence {sn} of completely pos-itive contractions from A to B which is dense among all such completely positivecontractions. We may assume that each sn occurs infinitely often in the sequence.Let F1 ⊆ F2 ⊆ F3 ⊆ · · · be a sequence of finite sets in A whose union is densein A and let K1 ⊆ K2 ⊆ K3 ⊆ · · · be a sequence of compact subsets of G whoseunion is G. For each pair n, l ∈ N, let Xn,l denote the set of pairs (w, π), wherew ∈ B, ‖w‖ ≤ 1, and π : M(A) → M(B) is a strictly continuous equivariant ∗-homomorphism such that ‖sn(a)−w∗π(a)w‖≤ 1/ l for a ∈ Fl . Note that Xn,l 6= ∅by Theorem 5.2. For each m ∈ N, set

δn,l,m = inf{

supg∈Km

‖g ·w−w‖ : (w, π) ∈ Xn,l}.

388 KLAUS THOMSEN

For each triple n,m, l ∈ N, choose a pair (wn,l,m, πn,l,m) ∈ Xn,l such that

supg∈Km

‖g ·wn,l,m −wn,l,m‖ ≤ δn,l,m +1l.

Let

π(x)=

∑n,l,m

Sn,l,mπn,l,m(x)S∗

n,l,m,

where {Sn,l,m} is a family of G-invariant isometries such that S∗

n,i,m Sk, j,l = 0 when(n, i,m) 6= (k, j, l) and

∑n,i,m Sn,i,m S∗

n,i,m = 1 in the strict topology.To show that the saturation of π |A is absorbing it suffices by Proposition 5.6 to

show that π |A has property (A) of that proposition, and to show that the saturationof π |A is equivariantly absorbing it suffices to show that π |A has property (1)of Theorem 5.4. We will establish these two properties of π |A simultaneously.Since π is strictly continuous this will complete the proof. Consider therefore acompletely positive contraction ϕ : A → M(B) such that either (a) ϕ(A) ⊆ B or(b) ϕ is equivariant. Let k ∈ N and ε > 0 be given. We will show that in both casesthere is an element W ∈ M(B) such that ‖W‖≤

√1 + ε and ‖ϕ(a)−W ∗π(a)W‖≤

3ε, a ∈ Fk . Then π |A will clearly satisfy condition (A). Furthermore, we will showthat in case (b) W can be chosen such that supg∈Kk

‖g · W − W‖ ≤ 2ε. Then π |A

will clearly satisfy condition (1) of Theorem 5.4, and we will be done.To simplify notation, set K = Kk and F = Fk . By using an asymptotically G-

invariant approximate unit for B which asymptotically commutes with the range ofϕ— for which see Lemma 5.1 — we can construct a sequence { fi } in B such that0 ≤ fi ≤ 1, ‖g · fi − fi‖< 2−iε for all i and g ∈ K ,

∑∞

i=1 f 2i = 1,

∑∞

i=1 fiϕ(a) fi

converge in the strict topology for all a ∈ A and∥∥∥∥ ∞∑i=1

fiϕ(x) fi −ϕ(x)∥∥∥∥≤ ε,(5–15)

for all x ∈ F . We claim that for any i ∈ N, any finite set H ⊆ A, any δ > 0 and anyN ∈ N, there is a pair n, l ∈ N such that n ≥ N , and

‖ fiϕ(a) fi −w∗

n,l,kπn,l,k(a)wn,l,k‖ ≤ δ,(5–16)

for all a ∈ H , and

‖g ·wn,l,k −wn,l,k‖ ≤ 2−iε,(5–17)

for all g ∈ K , in case (b). To see this observe first that in case (a) Theorem 5.2 givesus a sequence {Wn} of contractions in M(B) and a strictly continuous equivariant∗-homomorphism π0 : M(A)→ M(B) such that limn→∞ W ∗

n π0(a)Wn = ϕ(a) forall a ∈ A. In case (b) Proposition 5.3 does the same, and more: In case (b) we can


choose {Wn} to be asymptotically G-invariant. We first choose m ∈ N so large that

(5–18)∥∥ϕ(a)− W ∗

mπ0(a)Wm∥∥≤

12δ

for all a ∈ H , and ‖g · (Wm fi )− Wm fi‖ ≤ 2−iε, g ∈ K , in case (b). Then wechoose l ∈ N so large that

(i) 2/ l < δ/6,

(ii) H ⊆δ/6 Fl , meaning that for all x ∈ H there is a y ∈ Fl such that ‖x −y‖≤ δ/6;

in case (b) we require that l satisfy the additional condition

(iii) 1l ≤ 2−iε− max{‖g · (Wm fi )− Wm fi‖ : g ∈ K }.

Then choose n ≥ N such that

‖sn(x)− fi W ∗

mπ0(x)Wm fi‖ ≤1l

(5–19)

for all x ∈ Fl . It follows from (5–19) that δn,l,k ≤ max{‖g · (Wm fi )− Wm fi‖ :

g ∈ K }, and hence by (iii) that ‖g ·wn,l,k −wn,l,k‖ ≤ δn,l,k + 1/ l ≤ 2−iε for allg ∈ K , provided of course that we are in case (b). Thus (5–17) holds in this case.Since ‖sn(x)−w∗

n,l,kπn,l,k(x)wn,l,k‖ ≤ 1/ l for x ∈ Fl , it follows from (5–19) that‖ fi W ∗

mπ0(x)Wm fi −w∗

n,l,kπn,l,k(x)wn,l,k‖ ≤ 2/ l for all x ∈ Fl . Combining thiswith i) and ii) we obtain that ‖ fi W ∗

mπ0(a)Wm fi −w∗

n,l,kπn,l,k(a)wn,l,k‖ ≤12δ for

all a ∈ H . In combination with (5–18) this gives us (5–16).For each i we choose an element ui ∈ A, 0 ≤ ui ≤ 1, such that

‖ui xui − x‖ ≤ 2−iε,(5–20)

x ∈ F , and ‖g · ui − ui‖ ≤ 2−iε, g ∈ K ; we can do this by Lemma 5.1. Let b be astrictly positive element of B. By using the property established above and the factthat πn,l,k( · ) = S∗

n,l,kπ( · )Sn,l,k , we see that we can choose a sequence l ′i , i ∈ N,of contractions in B such that

(iv) l ′i∗π(A)l ′j = {0}, i 6= j ,

(v) ‖l ′i∗π(ui )b‖ ≤ 2−i ,

(vi) ‖l ′i∗π(x)l ′i − fiϕ(x) fi‖ ≤ 2−iε, x ∈ {u2

i } ∪ ui Fui ,

and in case (b),

(vii) ‖g · l ′i − l ′i‖ ≤ 2−iε, g ∈ K .

390 KLAUS THOMSEN

Set li = π(ui )l ′i . We claim that∑

∞

i=1 li converges in the strict topology of M(B).Since∥∥∥∥ m∑

i=n

li b∥∥∥∥2

≤ ‖b‖

∥∥∥∥ m∑i, j=n

l∗i l j b∥∥∥∥= ‖b‖

∥∥∥∥ m∑i=n

l∗i li b∥∥∥∥

= ‖b‖

∥∥∥∥ m∑i=n

l ′i∗π(u2

i )l′

i b∥∥∥∥ (by (iv))

≤ ε‖b‖

m∑i=n

2−i+ ‖b‖

∥∥∥∥ m∑i=n

fiϕ(u2i ) fi b

∥∥∥∥ (by (vi))

≤ ε‖b‖

m∑i=n

2−i+ ‖b‖

√√√√∥∥∥∥b( m∑

i=n

fiϕ(u2i ) fi

)2

b∥∥∥∥

≤ ε‖b‖

m∑i=n

2−i+ ‖b‖

√√√√∥∥∥∥b( m∑

i=n

f 2i

)b∥∥∥∥,

we see that∑

∞

i=1 li b converges in B. Since ‖l∗i b‖≤2−i by (v),∑

∞

i=1 l∗i b convergesalso in B. Since b is a strictly positive element, and since

supm

∥∥∥∥ m∑i=1

li

∥∥∥∥≤√

1 + ε

by (iv) and (vi), it follows that∑

∞

i=1 li converges in the strict topology to an elementW of M(B) whose norm does not exceed

√1 + ε. For x ∈ F we have

∥∥W ∗π(x)W−ϕ(x)∥∥≤ ε+

∥∥∥∥W ∗π(x)W −

∞∑i=1

fiϕ(x) fi

∥∥∥∥ (by (5–15))

= ε+

∥∥∥∥ ∞∑i=1

l ′i∗π(ui xui )l ′i −

∞∑i=1

fiϕ(x) fi

∥∥∥∥ (by (iv))

≤ 2ε+

∥∥∥∥ ∞∑i=1

l ′i∗π(ui xui )l ′i −

∞∑i=1

fiϕ(ui xui ) fi

∥∥∥∥ (by (5–20))

≤ 3ε (by (vi)).

Finally,

‖g · li − li‖ ≤ ‖g · l ′i − l ′i‖ +‖g · ui − ui‖ ≤ 2−i+1ε, g ∈ K ,

by (vii) and the choice of ui , in case (b). Hence ‖g · W − W‖ ≤ 2ε for all g ∈ K ,in case (b). �


6. Duality in equivariant KK -theory

In this section we combine the results of Sections 3 and 4 with those of Section 5.In this way we obtain the duality results for equivariant KK -theory relativelypainlessly.

We assume now that A and B are separable G-algebras, both G-stable.

Lemma 6.1. Let A and B be G-algebras, B stable. Let π : A → M(B) be asaturated equivariant ∗-homomorphism and set

E = {m ∈ M(B)G : mπ(a)= π(a)m, a ∈ A}.

Then K∗(E)= {0}.

Proof. The argument is well-known so we will be sketchy. Since π is saturated,E contains a sequence {Vi } of G-invariant isometries with orthogonal ranges suchthat

∑∞

i=1 Vi V ∗

i = 1 in the strict topology. We can then define a ∗-homomorphismψ : E → E such that ψ(m)=

∑∞

i=2 Vi mV ∗

i , where the sum converges in the stricttopology. Then ψ ⊕ idE = Ad U ◦ψ for a unitary U ∈ E , and we conclude thatψ∗ + id = ψ∗ in K -theory. It follows that K∗(E)= {0}. �

Given an equivariantly absorbing ∗-homomorphism π : A → M(B), set

Aπ = {x ∈ M(B) : xπ(a)−π(a)x ∈ B , a ∈ A},

andBπ = {x ∈ Aπ : xπ(A)⊆ B}.

Then Bπ is a closed two-sided ideal in Aπ and we set Dπ = Aπ/Bπ . Note that Gacts by automorphisms on Aπ which leave Bπ globally invariant, and we get anaction of G on Dπ . None of these actions are continuous in general.

If τ : A → M(B) is another equivariantly absorbing ∗-homomorphism, there is aunitaryw∈ M(B) such that Adw◦π(a)−τ(a)∈ B for all a ∈ A and g·w−w∈ B forall g ∈ G; see Proposition 5.5. It follows that there is an equivariant ∗-isomorphismbetween Dπ and Dτ . In particular, it follows that DG

π ' DGτ .

Let u be a unitary in Mn(DGπ ). Choose v ∈ Mn(Aπ ) such that idMn ⊗q(v)= u.

Define πn: A → LB(Bn) by

πn(a)(b1, b2, . . . , bn)= (π(a)b1, π(a)b2, . . . , π(a)bn) .

Let Bn⊕ Bn be graded by (x, y) 7→ (x,−y). Then

(6–1)(Bn

⊕ Bn,(πn

πn

),(v∗

v))

is an even Kasparov triple for A and B. It is easy to see that the class of (6–1)in KK 0

G(A, B) only depends on the class of u in K1(DGπ ). Thus the construc-

tion gives rise to a map 2 : K1(DGπ ) → KK 0

G(A, B) which is easily seen to be ahomomorphism.

392 KLAUS THOMSEN

In the following we will let 1m and 0m denote the unit and the zero element ofMm(M(B)), respectively. We will identify Mm(M(B)) and M(Mm(B)).

Theorem 6.2. Let A and B be separable G-algebras, both G-stable. Then 2 :

K1(DGπ )→ KK 0

G(A, B) is an isomorphism.

Proof. 2 is injective: Let u ∈ Mn(DGπ ) be a unitary and choose v ∈ Mn(Aπ ) such

that idMn ⊗q(v)= u. Assume that[Bn

⊕ Bn,(πn

πn

),(v∗

v)]

= 0 in KK 0G(A, B).

By Theorem 3.10 this means that there are elementary degenerate even Kasparovtriples D1 and D2 such that

(Bn

⊕ Bn,(πn

πn

),(v∗

v))

⊕D1 is operator homotopicto(Bn

⊕ Bn,(πn

πn

),(

11))

⊕ D2. Since D1 and D2 are degenerate we can definea new degenerate even Kasparov A − B-module D by

D = 0 ⊕ D1 ⊕ D2 ⊕ D1 ⊕ D2 ⊕ D1 ⊕ D2 ⊕ · · · .

Then D1 ⊕ D and D2 ⊕ D are both isomorphic to D and hence(Bn

⊕ Bn,(πn

πn

),(v∗

v))

⊕ D

is operator homotopic to(Bn

⊕ Bn,(πn

πn

),(

11))

⊕D. Note that D is isomorphicto an elementary even Kasparov triple since D1 and D2 are. Up to isomorphism, itmust therefore have the form

D =(Be,

(λ+

λ−

),(

ba ))

,

where λ± : A → M(B) are saturated equivariant ∗-homomorphisms and a, b ∈

M(B). By performing the same alterations to D as was performed to E in [Jensenand Thomsen 1991, pages 125–126] we may assume that a = w and b = w∗ forsome unitary w ∈ M(B). Note that wλ−(a)w∗

= λ+(a), (g ·w)λ±(a)= wλ±(a)for all a ∈ A and all g ∈ G since D is degenerate. By adding to D the sum(

Be,(λ−

λ+

),(ww∗ ))

⊕(Be,

(ππ

),(

11))

we may assume that λ+ = λ− and that λ = λ+ = λ− is equivariantly absorbing.Now the operator homotopy between(

Bn⊕ Bn,

(πn

πn

),(v∗

v))

⊕(Be,

(λλ

),(w∗

w))

and (Bn

⊕ Bn,(πn

πn

),(

11))

⊕(Be,

(λλ

),(w∗

w))

gives us a G-invariant unitary S ∈ Mn+1 (M(B)) such that

S(πn(a)

λ(a)

)=(πn(a)

λ(a)

)S

for all a ∈ A, and a norm-continuous path Ft , t ∈ [0, 1], in Mn+1(M(B)) suchthat F0S = S

(1nw

), F1 =

(vw

), and the following inclusions are satisfied for all


values of g, t, a: (Ft F∗

t − 1n+1) (

πn(a)λ(a)

)∈ Mn+1(B),(

F∗

t Ft − 1n+1) (

πn(a)λ(a)

)∈ Mn+1(B),

(g · Ft − Ft)(πn(a)

λ(a)

)∈ Mn+1(B),

and

Ft(πn(a)

λ(a)

)−(πn(a)

λ(a)

)Ft ∈ Mn+1(B).

As an equivariant ∗-homomorphism A → Mn+1(M(B)), ν =(πn

λ

)is saturated,

since π and λ are. By Lemma 6.1 we can therefore find an m ∈ N and a norm-continuous path of unitaries in

{x ∈ Mm(n+1)(M(B)G) : xνm(a)= νm(a)x, a ∈ A}

connecting( S

1(m−1)(n+1)

)to 1m(n+1). In combination with F this gives us a norm-

continuous path Ht , t ∈ [0, 1], in Mm(n+1)(M(B)G

)such that

H0 =

1n

w

1(m−1)(n+1)

, H1 =

v w1(m−1)(n+1)

,and moreover the following elements lie in Mm(n+1)(B):

(Ht H∗

t − 1m(n+1))νm(a),(

H∗t Ht − 1m(n+1)

)νm(a), (g · Ht − Ht) ν

m(a) and Htνm(a)−νm(a)Ht , for all t , g,

and a. Since λ and π both are equivariantly absorbing there is a unitaryw0 ∈ M(B)such that g ·w0 −w0 ∈ B for all g ∈ G and w0λ(a)w∗

0 − π(a) ∈ B, a ∈ A; seeProposition 5.5. Set

W = diag(1n, w0, 1n, w0, . . . , 1n, w0︸︷︷︸

m times

)∈ Mm(n+1)(M(B))

andG t = W Ht W ∗.

Then G t is a norm-continuous path in Mm(n+1)(M(B)) such that

G0 =

1n

w0ww∗

01(m−1)(n+1)

, G1 =

v w0ww∗

01(m−1)(n+1)

,and moreover the following elements lie in Mm(n+1)(B), again for all t , g, and a:(G t G∗

t −1m(n+1)πm(n+1)(a), (G∗

t G t −1m(n+1)πm(n+1)(a), (g ·G t −G tπ

m(n+1)(a)and G tπ

m(n+1)(a)−πm(n+1)(a)G t . Thus(idMm(n+1) ⊗q

)(G t) is a path of unitaries

394 KLAUS THOMSEN

in Mm(n+1)(DGπ

)connectingu

q(w0ww∗

0)

1(m−1)(n+1)

to

1n

q(w0ww∗

0)

1(m−1)(n+1)

.2 is surjective: Let (E, ψ, F) be an even Kasparov triple for A and B. By Theorem3.10 we may assume that E = Be. By Lemma 3.2 we may assume that ψ =

(ϕ, ϕ). The constructions in [Jensen and Thomsen 1991, pages 125–126] showthat [Be, ψ, F] ∈ KKG(A, B) is also represented by a Kasparov triple of the form(

Be,

(ϕ

ϕ

),

(v

v∗

)),

where ϕ : A → M(B) is an equivariant ∗-homomorphism and v∈ M(B) is a unitary.By adding on (

Be,

(π

π

),

(1

1

)),

and using that π is equivariantly absorbing we may assume that there is a unitaryu ∈ M(B) such that g·u−u ∈ B, g ∈ G, and uϕ(a)u∗

−π(a)∈ B for all a ∈ A. Let Xbe the graded Hilbert B,G-module which as a graded Hilbert B-module is B ⊕ B,but with the representation of G changed to g ·(b1, b2)=

(uβg (u∗b1) , uβg (u∗b2)

).

Then(Be,

(ϕϕ

),(v∗

v))

is isomorphic to(X,(

Ad u ◦ϕ

Ad u ◦ϕ

),

(uvu∗

uv∗u∗

)).

Thanks to the properties of u a rotation argument works to show that(X,(

Ad u ◦ϕ

Ad u ◦ϕ

),

(uvu∗

uv∗u∗

))⊕

(Be,

(π

π

),

(1

1

))is operator homotopic to(

X,(

Ad u ◦ϕ

Ad u ◦ϕ

),

(1

1

))⊕

(Be,

(π

π

),

(uvu∗

uv∗u∗

)).

Thus uvu∗ is a unitary in Aπ such that q(uvu∗) ∈ DGπ and 2([q (uvu∗)]) =

[E, ψ, F] in KK 0G(A, B). �

Let next Q be a projection in Mn(DGπ

), and let P ∈ Mn (Aπ ) be a lift of P .

Then (Bn, πn, Q) is an odd Kasparov for A and B and we can define a map 20 :

K0(DGπ

)→ KK 1

G(A, B) such that 20[Q] = [Bn, πn, Q]. By using Theorem 4.3in place of Theorem 3.10, we can adopt the previous proof to obtain:


Theorem 6.3. Let A and B be separable G-algebras, both G-stable. Then 20 :

K0(DGπ )→ KK 1

G(A, B) is an isomorphism.

Lemma 6.4. Let A and B be separable G-algebras, both G-stable. Let π :

M(A)→ M(B) be a strictly continuous ∗-homomorphism such that π |A is equiv-ariantly absorbing. It follows that q

(AGπ

)=DG

π , and that q∗ : K∗

(AGπ

)→ K∗

(DGπ

)is an isomorphism.

Proof. Set E = π(1) which is a G-invariant projection in M(B) such that E B =

π(A)B. Let x ∈ DGπ be self-adjoint and choose a self-adjoint y ∈ Aπ such q(y)= x .

By applying Lemma 3.3 to the triple (B, π, y) we see that there is a G-invariantelement z ∈ LB(E B) such that yπ(a)−π(a)z ∈ KB(E B, B) and π(a)y − zπ(a) ∈KB(B, E B) for all a ∈ A. Define z0 ∈ M(B) such that z0b = zEb for all b ∈ B.Then z0 ∈ M(B)G and yπ(a)− π(a)z0 ∈ B, π(a)y − z0π(a) ∈ B for all a ∈ A.It follows that z0π(a)− π(a)z0 ∈ B and π(a) (y − z0) ∈ B for all a ∈ A so thatz0 ∈ AG

π and q(z0)= q(y)= x . Hence q(AGπ

)= DG

π .It suffices now to show that K∗

(BGπ

)= 0. To this end, consider the C∗-algebra

Y ={

X ∈ M2(M(B)G) : X(π(a)

0

),(π(a)

0

)X ∈ M2(B), a ∈ A

}.

Define κ : BGπ → Y by κ(x) =

( x0

). We claim that κ∗ : K∗

(AGπ

)→ K∗ (Y) is

injective and that κ∗ = 0. To prove injectivity of κ∗, note that since π absorbs 0there is a G-invariant unitary U ∈ LB(B ⊕ B, B) such that

(6–2) U(π(a)

0

)U∗

−π(a) ∈ B

for all a ∈ B. Then X 7→ U XU∗ is an equivariant ∗-homomorphism λ : Y → BGπ .

Define V : B → B by V b = U (b, 0), and observe that V is adjointable withadjoint V ∗

: B → B given by p1U∗b, where p1 : B ⊕ B → B is the projectionto the first coordinate. V is then a G-invariant isometry V ∈ M(B)G such thatλ ◦ κ = Ad U ◦ κ = Ad V , and Vπ(a)V ∗

= U(π(a)

0

)U∗. It follows from the last

equality and (6–2) that Vπ(a)−π(a)V ∈ B for all a ∈ V , and then that xV ∈ BGπ

when x ∈ BGπ . Therefore V is an isometry in M

(BGπ

)and hence (Ad V )∗ = id in

K -theory. Consequently λ∗ ◦ κ∗ = id in K -theory, and κ∗ must be injective. Onthe other hand, observe that κ is homotopic via a standard rotation argument tothe ∗-homomorphism x 7→

( 0x

), which factors through M(B)G . Since the zero

homomorphism is saturated, it follows from Lemma 6.1 that K∗

(M(B)G

)= 0.

Thus κ∗ = 0. �

Theorem 6.5. Let A and B be separable G-algebras, both G-stable. Let π :

M(A)→ M(B) be a strictly continuous ∗-homomorphism such that π |A is equiv-ariantly absorbing. It follows that 2 ◦ q∗ : K1

(AGπ

)→ KK 0

G(A, B) and 20 ◦ q∗ :

K0(AGπ

)→ KK 1

G(A, B) are both isomorphisms.

Proof. Combine Theorems 6.2 and 6.3 with Lemma 6.4. �

396 KLAUS THOMSEN

By Theorem 5.7 we can choose the ∗-homomorphism π of Theorem 6.5 suchthat π |A : A → M(B) is absorbing (nonequivariantly), and it follows then from[Thomsen 2001] that the K -theory of Aπ gives us the nonequivariant KK -groupsKK i (A, B), i = 0, 1. Under these identifications the canonical forgetful maps

KK iG (A, B)→ KK i (A, B) ,

i =0, 1, become the maps Ki(AGπ

)→ Ki (Aπ ), i =0, 1, induced by the embedding

AGπ ⊆ Aπ .

References

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[Lance 1995] E. C. Lance, Hilbert C∗-modules, London Mathematical Society Lecture Note Series210, Cambridge University Press, Cambridge, 1995. MR 96k:46100 Zbl 0288.46054

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Received March 4, 2004. Revised March 18, 2005.

KLAUS THOMSEN

INSTITUT FOR MATEMATISKE FAG

NY MUNKEGADE

8000 ARHUS CDENMARK

[email protected]

CONTENTS

Volume 222, no. 1 and no. 2

Irene I. Bouw with Stefan Wewers 185

Henrique Bursztyn and Stefan Waldmann: Completely positive inner products andstrong Morita equivalence 201

Roberto Camporesi: A generalization of the Cartan–Helgason theorem forRiemannian symmetric spaces of rank one 1

Wen-Chen Chi, King Fai Lai and Ki-Seng Tan: Integer points on elliptic curves 237

Manuel D. Contreras and Santiago Dıaz-Madrigal: Analytic flows on the unit disk:angular derivatives and boundary fixed points 253

Kenneth R. Davidson and Ronald G. Douglas: The generalized Berezin transformand commutator ideals 29

Santiago Díaz-Madrigal with Manuel D. Contreras 253

Ronald G. Douglas with Kenneth R. Davidson 29

Dennis Garity, Dusan Repovs and Matjaz Zeljko: Uncountably many inequivalentLipschitz homogeneous Cantor sets in R3 287

Włodzimierz Jelonek: Bihermitian Gray surfaces 57

Ming-chang Kang: Noether’s problem for dihedral 2-groups II 301

King Fai Lai with Wen-Chen Chi and Ki-Seng Tan 237

Bertrand Lemaire: Intégrabilité locale des caractères de SLn(D) 69

Goran Muic: Reducibility of standard representations 133

Cam Van Quach Hongler and Claude Weber: A Murasugi decomposition for achiralalternating links 317

Dusan Repovš with Dennis Garity and Matjaz Zeljko 287

Rob Schneiderman: Simple Whitney towers, half-gropes and the Arf invariant of aknot 169

Ki-Seng Tan with Wen-Chen Chi and King Fai Lai 237

Terence Tao: A new bound for finite field Besicovitch sets in four dimensions 337

Klaus Thomsen: Duality in equivariant KK -theory 365

Stefan Waldmann with Henrique Bursztyn 201

400

Claude Weber with Cam Van Quach Hongler 317

Stefan Wewers and Irene I. Bouw: Alternating groups as monodromy groups inpositive characteristic 185

Matjaz Željko with Dennis Garity and Dusan Repovs 287

Guidelines for Authors

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By submitting a manuscript you assert that it is original and is not under considerationfor publication elsewhere. Instructions on manuscript preparation are provided below. Forfurther information, visit the web address above or write to [email protected] orto Pacific Journal of Mathematics, University of California, Los Angeles, CA 90095–1555.Correspondence by email is requested for convenience and speed.

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Completely positive inner products and strong Morita equivalence 201HENRIQUE BURSZTYN AND STEFAN WALDMANN

Integer points on elliptic curves 237WEN-CHEN CHI, KING FAI LAI AND KI-SENG TAN

Analytic flows on the unit disk: angular derivatives and boundary fixed points 253MANUEL D. CONTRERAS AND SANTIAGO DÍAZ-MADRIGAL

Uncountably many inequivalent Lipschitz homogeneous Cantor sets in R3 287DENNIS GARITY, DUŠAN REPOVŠ AND MATJAŽ ŽELJKO

Noether’s problem for dihedral 2-groups II 301MING-CHANG KANG

A Murasugi decomposition for achiral alternating links 317CAM VAN QUACH HONGLER AND CLAUDE WEBER

A new bound for finite field Besicovitch sets in four dimensions 337TERENCE TAO

Duality in equivariant KK -theory 365KLAUS THOMSEN

0030-8730(200512)222:2;1-K

PacificJournalofM

athematics

2005Vol.222,N

o.2

PacificJournal ofMathematics


msp · pacific journal of mathematics vol. 222, no. 2, 2005 completely positive inner products and...

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