Fields Institute Summer School Operator Algebras – August 2007 – Ottawa

Discrete quantum groups and their probabilistic boundaries

Stefaan Vaes, K.U.Leuven, stefaan.vaes@wis.kuleuven.be

Disclaimer. Due to lack of time, these lecture notes are in very premature form and probably con-tain a lot of errors and misprints. I would be glad to hear any of your questions/comments/suggestions.

Exercises and discussions. I included several exercises in the text. They are not meant to bevery deep, but rather to train you and to let you understand how things work. If some people areinterested in asking questions or discussing the exercises, this can be organized in the afternoon.Please come and see me.

Some references. The following is by no means an exhaustive list of references, but rathera guide to enter the recent literature on compact or discrete quantum groups from the point ofview taken in the course.

References

[1] A very good beginners’ introduction to the theory of compact and discrete quantum groups.A. Maes & A. Van Daele, Notes on compact quantum groups. Nieuw Arch. Wisk. (4) 16 (1998),73–112. arXiv:math/9803122

[2] The paper where it all started from, published in 1998, but written in the beginning of the 1990’s.S.L. Woronowicz, Compact quantum groups. In Symetries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 845–884.Available on http://www.fuw.edu.pl/∼slworono/Prace.html

[3] Computation of the representation theory of the compact quantum group Ao(F ).T. Banica, Theorie des representations du groupe quantique compact libre O(n). C. R. Acad. Sci.Paris Ser. I Math. 322 (1996), 241–244.Available on http://picard.ups-tlse.fr/∼banica/pub.html

[4] Computation of the representation theory of the compact quantum group Au(F ).T. Banica, Le groupe quantique compact libre U(n). Commun. Math. Phys. 190 (1997), 143–172.Available on http://picard.ups-tlse.fr/∼banica/pub.html

[5] Construction of ergodic actions of the quantum groups SUq(2), Ao(F ) and Au(F ) with remarkableproperties. The main tool is the construction of a link C∗-algebra between monoidally equivalent quan-tum groups.J. Bichon, A. De Rijdt & S. Vaes, Ergodic coactions with large multiplicity and monoidal equiv-alence of quantum groups. Comm. Math. Phys. 262 (2006), 703–728. arXiv:math/0502018

[6] The tool of [5] is used to describe how the Poisson boundary behaves when passing to a monoidallyequivalent quantum group. In combination with [9], this allows to compute the Poisson boundary ofa large class of quantum groups, including the duals of Ao(F ).A. De Rijdt & N. Vander Vennet, Actions of monoidally equivalent compact quantum groupsand applications to probabilistic boundaries. Preprint. arXiv:math/0611175

[7] The first appearence of random walks and their Poisson boundaries in quantum group theory, moti-vated by the search for minimal actions and their relations to subfactors. The actual computation inthe paper of the Poisson boundary of the dual of SUq(2) has since then be simplified in [8] and even

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more in [9] recently.M. Izumi, Non-commutative Poisson boundaries and compact quantum group actions. Adv. Math.169 (2002), 1–57.

[8] The paper computes in a very elegant way what the title says. Since then, a simpler approach becameavailable in [9].M. Izumi, S. Neshveyev & L. Tuset, Poisson boundary of the dual of SUq(n). Comm. Math. Phys.262 (2006), 505-531. arXiv:math/0402074

[9] Computation of the Poisson boundary for all coamenable compact quantum groups having commuta-tive fusion rules. This includes all q-deformations of compact Lie groups and as such, generalizes andsimplifies the computations of [7] and [8].R. Tomatsu, A characterization of right coideals of quotient type and its application to classificationof Poisson boundaries. Preprint. arXiv:math/0611327

[10] A systematic study of minimal/outer actions of quantum groups and their relations with subfactors.S. Vaes, Strictly outer actions of groups and quantum groups. J. Reine Angew. Math. 578 (2005),147–184. arXiv:math/0211272

[11] Study of the compact quantum group Ao(F ) from an operator algebra point of view : we are interestedin the von Neumann algebra L∞(G). In certain cases, we obtain prime factors. The main tool is theconstruction of a Gromov boundary, in order to prove the Akemann-Ostrand property and to deduceOzawa’s solidity.S. Vaes & R.Vergnioux, The boundary of universal discrete quantum groups, exactness and facto-riality. Duke Math. J., to appear. arXiv:math/0509706

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1 Introduction

Topic

Discrete quantum groups Compact quantum groups

Operator algebras

Duality

Motivation

Discrete/compact quantum groups arise as follows :

• generalized symmetry :

– actions on operator algebras→ minimal actions, ergodic actions,→ random walks, Poisson boundaries,

– non-commutative geometry,

• subfactor theory.

Several of these aspects will arise during the course.

Important notational convention. The symbol ⊗ denotes the minimal tensor product of C∗-algebras as well as the tensor product of Hilbert spaces. The tensor product of von Neumannalgebras is denoted by ⊗, while the algebraic tensor product of vector spaces is denoted by ⊗alg.

2 Definition of a compact quantum group

Definition 2.1 (Woronowicz [2]). A compact quantum group G is a pair (C(G),∆) consisting of

• a unital C∗-algebra C(G),

• a unital ∗-homomorphism ∆ : C(G) → C(G)⊗ C(G),

satisfying

• co-associativity :(∆⊗ id)∆ = (id⊗∆)∆ , (2.1)

• left and right cancelation properties :

∆(C(G))(1⊗ C(G)) and ∆(C(G))(C(G)⊗ 1) (2.2)

are total in C(G)⊗ C(G).

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Where does this definition come from.

• Let A be an abelian C∗-algebra and ∆ : A→ A⊗A a unital ∗-homomorphism satisfying (2.1).→ compact semi-group G such that A = C(G) and ∆ : C(G) → C(G × G) : ∆(F )(x, y) =F (xy) for all x, y ∈ G. (Note that C(G)⊗ C(G) = C(G×G).)

• Then, (2.2) is equivalent with the left and right cancelation properties.

Proof. Suppose that ∆(C(G))(1 ⊗ C(G)) is total in C(G × G). This means that functionsof the form (x, y) 7→ F (xy)G(y) densely span C(G × G). If then x0y = x1y, it follows thatH(x0, y) = H(x1, y) for all H ∈ C(G×G), implying that x0 = x1.

Exercise 1. Prove the converse.

• A compact semi-group with left and right cancelation properties is a compact group.

Proof. We have to prove the existence of a unit element and of inverses as well as the continuityof the inverse operation.

Given g ∈ G, we can switch to the commutative compact semi-group generated by g andassume that G is commutative.

The closed ideals of G have the finite intersection property. So, take the smallest non-emptyclosed ideal I ⊂ G. Let g ∈ I. Then, gI ⊂ I so that by minimality, gI = I. Take e ∈ Isatisfying ge = g. Then, for all h ∈ G, gh = geh and by cancelation, h = eh for all h ∈ G.Then, keh = kh and hence, ke = k for all k ∈ G. We have found the unit element.

Moreover, e ∈ I and hence, G = eG ⊂ I. So, every closed non-empty ideal in G equals G. Inparticular, gG = G for all g ∈ G, proving the existence of an inverse for every g ∈ G.

Continuity of the inverse operation follows from its closedness.

Conclusion : compact quantum groups G with abelian underlying C∗-algebra C(G) correspond tocompact groups.→ reason for the notation C(G), although in general, there is no underlying space G.

Example 2.2. Let Γ be a discrete group. Define G = Γ as

• C(G) is either C∗(Γ) or C∗r (Γ),

• ∆(ug) = ug ⊗ ug for all g ∈ Γ.

Later : many naturally appearing examples of compact quantum groups.

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3 Existence and uniqueness of the Haar measure

Recall.

• A compact group G admits a unique translation-invariant probability measure, called theHaar measure.

• Probability measures on a compact space X correspond to states on the C∗-algebra C(X).

• The state space of a unital C∗-algebra is compact in the weak∗-topology.

Theorem 3.1 (Woronowicz [2], proof taken from [1]). Let G = (C(G),∆) be a compact quantumgroup. Then, C(G) admits a unique state h satisfying

(id⊗ h)∆(a) = h(a)1 = (h⊗ id)∆(a) for all a ∈ C(G) .

We call h the Haar state of G.

Preparation. The convolution product of two states ω, ρ ∈ C(G)∗ is given by

ω ∗ ρ = (ω ⊗ ρ)∆ .

Exercise 2. Convince yourself that this definition of the convolution product, coincides with theusual convolution product of measures in the case G is an ordinary compact group.

We have to prove the existence of a state h satisfying ω ∗ h = h = h ∗ ω for all states ω.

Proof of Theorem 3.1.Step 1. If ω is a state on C(G), there exists a state h on C(G) satisfying ω ∗ h = h = h ∗ ω.It suffices to take a weak∗-accumulation point of the sequence of states given by

1n

n∑k=1

ω∗k .

Step 2. If ω and h are states on C(G) satisfying ω ∗ h = h = h ∗ ω, then

ρ ∗ h = ρ(1)h = h ∗ ρ whenever 0 ≤ ρ ≤ ω .

Choose a ∈ C(G) arbitrary. Put b = (id⊗ h)∆(a). Compute

(id⊗ ω)∆(b) = (id⊗ (ω ∗ h))∆(a) = (id⊗ h)∆(a) = b .

It follows that(h⊗ ω)

((∆(b)− b⊗ 1)∗(∆(b)− b⊗ 1)

)= 0 .

But then also(h⊗ ρ)

((∆(b)− b⊗ 1)∗(∆(b)− b⊗ 1)

)= 0 .

Using the Cauchy-Schwartz inequality, we get

(h⊗ ρ)((c⊗ 1)(∆(b)− b⊗ 1)

)= 0 for all c ∈ C(G) .

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Finally compute

(h⊗ (ρ ∗ h))((c⊗ 1)∆(a)

)= (h⊗ ρ⊗ h)

((c⊗ 1⊗ 1)(id⊗∆)∆(a)

)= (h⊗ ρ)

((c⊗ 1)∆(b)

)= h(cb) ρ(1) = ρ(1)(h⊗ h)

((c⊗ 1)∆(a)

).

By (2.2), we get

(h⊗ (ρ ∗ h))(d) = ρ(1)(h⊗ h)(d) for all d ∈ C(G)⊗ C(G) .

This means that ρ ∗ h = ρ(1)h.

End of the proof. If ω1, . . . , ωk are states on C(G), put ω = 1k (ω1 + · · · + ωk). Combining Steps

1 and 2, we find a state h on C(G) such that ωi ∗ h = h = h ∗ ωi for all i = 1, . . . , k. The finiteintersection property yields a state h on C(G) satisfying ω ∗ h = h = h ∗ ω for all states ω.

4 Unitary representations of compact quantum groups

4.1 Some reminders about the representation theory of compact groups

We only consider unitary, strongly continuous representations U : G→ U(H).

• Every irreducible representation is finite dimensional.

• Every representation is a direct sum of irreducible representations.

• The regular representation G → U(L2(G)) contains every irreducible representation. Moreprecisely, the regular representation contains exactly dim(U) times the irreducible represen-tation U .

• Given representations U : G → U(H) and V : G → U(K), we have the tensor productrepresentation

U ⊗ V : G→ U(H ⊗K) : (U ⊗ V )(g) = U(g)⊗ V (g) .

Example 4.1. Take the compact group G = SU(2). We have two obvious representations : thefundamental representation on the Hilbert space C2 and the trivial representation on the Hilbertspace C.

First take the n-fold tensor power of the fundamental representation, which acts on the Hilbert space(C2)⊗n. On the same Hilbert space (C2)⊗n acts the symmetric group Sn. Denote by Symn(C2) thesubspace of (C2)⊗n consisting of Sn-invariant vectors.

Exercise 3. Check that the representations of Sn and SU(2) on (C2)⊗n commute, so that Symn(C2)is globally invariant under SU(2).

Fact (but not entirely trivial to prove). The irreducible representations of SU(2) are exactlygiven (up to unitary equivalence) by the representations on Symn(C2), n ∈ N. (With the conventionthat n = 0 corresponds to the trivial representation and n = 1 to the fundamental representation.)

Denote by Un the representation on Symn(C2). Then,

Un ⊗ Um∼= U|n−m| ⊕ U|n−m|+2 ⊕ · · · ⊕ Un+m . (4.1)

In general, the tensor product of two irreducible representations decomposes as a direct sum ofirreducible representations. The rules describing this decomposition (as e.g. concretely done in(4.1) for SU(2)) are called the fusion rules of the compact group.

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4.2 Representations of compact quantum groups

Definition 4.2. A unitary representation of a compact quantum group G on a (finite-dimensional)Hilbert space H is a unitary operator U ∈ B(H)⊗ C(G) satisfying

(id⊗∆)(U) = U12 U13 .

The notation U12, U13. They denote the obvious elements in B(H)⊗ C(G)⊗ C(G), with U12 =U ⊗ 1 and U13 = (id⊗ σ)(U ⊗ 1), where σ : C(G)⊗C(G) → C(G)⊗C(G) flips the tensor factors.

Representations on infinite dimensional Hilbert spaces :replace B(H) ⊗ C(G) by M(K(H) ⊗ C(G)) = L(H ⊗ C(G)), where L(H ⊗ C(G)) denotes theC∗-algebra of adjointable operators on the Hilbert C∗-module H ⊗ C(G).

Choosing an orthonormal basis in H, an n-dimensional unitary representation of G is a unitarymatrix U ∈ Mn(C(G)) satisfying

∆(Uij) =n∑

k=1

Uik ⊗ Ukj .

Exercise 4. Check that this equation is indeed equivalent with the one in 4.2.

Definition 4.3. Let U1 and U2 be unitary representations of G on H1,H2. We define

Mor(U1, U2) = {T ∈ B(H2,H1) | U1(T ⊗ 1) = (T ⊗ 1)U2} .

The elements of Mor(U1, U2) are called intertwiners between U1 and U2.

We denote End(U) := Mor(U,U). A unitary representation U is called irreducible if End(U) = C1.Equivalently, U is irreducible if U cannot be written as the direct sum of two non-zero representa-tions.

Theorem 4.4 (Woronowicz, [2]). Let G be a compact quantum group. Every unitary representationis the direct sum of irreducible representations. Every irreducible unitary representation is finite-dimensional.

Proof. Let U ∈ M(K(H)⊗ C(G)) be a unitary representation of G.

Exercise 1. Convince yourself of the fact that End(U) is a C∗-algebra. Argue that it is thensufficient to prove that the C∗-algebra A := End(U)∩K(H) acts non-degenerately on H (meaningthat AH is total in H).

Exercise 2. Let S ∈ B(H) be arbitrary. Prove that T := (id⊗h)(U∗(S⊗1)U) belongs to End(U).Note that T ∈ End(U) ∩ K(H) whenever S ∈ K(H).

End of the proof. If End(U) ∩ K(H) acts degenerately, we find a non-zero vector ξ ∈ H suchthat

(id⊗ h)(U∗(S ⊗ 1)U)ξ = 0 for all S ∈ K(H) .

But then, the same is true for all S ∈ B(H), in particular for S = 1, contradiction.

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4.3 Direct sum and tensor product of representations

Given representations U on H and V on K, it is quite obvious how to define the direct sumrepresentation U ⊕ V on H ⊕K.

The tensor product representation U T© V is defined by the formula

U T© V := U13 V23 .

Note that U T© V ∈ M(K(H ⊗K)⊗ C(G)) is it should.

Exercise 5. Prove that U T© V is again a representation.

Denote by Σ : H ⊗K → K ⊗H the flip map. Then, U T© V and (Σ∗ ⊗ 1)(V T© U)(Σ⊗ 1) are bothunitary representations on H ⊗K. But since in general, the C∗-algebra C(G) is non-commutative,these two representations are not necessarily the same. They even do not need to be unitarilyequivalent.

If U T©V and V T©U are unitarily equivalent for all U and V (which does not mean that it is the flipmap making them unitarily equivalent), the quantum group G is said to have commutative fusionrules. We will see below examples of compact quantum groups with C(G) non-commutative, butyet with commutative fusion rules.

4.4 The contragredient of a unitary representation, orthogonality relations

If G is a compact group and U : G→ U(H) a unitary representation, one defines the contragredientrepresentation U on the dual Hilbert space H as

U(g) ξ := U(g)ξ .

Problem when dealing with compact quantum groups : the same procedure does not necessarilyyield a unitary representation. But it always yields a unitarizable representation.

All of the following are related and discussed below :

→ unitarizing the contragredient representation,

→ the Haar state h on C(G) is not a trace, but ... ,

→ orthogonality relations.

Let H be a finite-dimensional Hilbert space, with dual Hilbert space H, consisting of the vectorsξ, ξ ∈ H with scalar product 〈ξ, η〉 = 〈η, ξ〉. For any C∗-algebra A, define the conjugation map

: B(H)⊗A→ B(H)⊗A : T ⊗ a = T ⊗ a∗ .

For non-commutative C∗-algebras A, the conjugation map has very bad properties. Nevertheless,you can check the following.

Exercise 6. Let U ∈ B(H) ⊗ C(G) be a finite-dimensional unitary representation of G. Then,U ∈ B(H)⊗ C(G) is again a representation, meaning that

(id⊗∆)(U) = U12 U13 ,

but U is not necessarily unitary.

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Although U is not necessarily unitary, it can be unitarized. We do not prove this crucial result,but refer to Section 6 in [1].

Theorem 4.5 (Woronowicz, [2]). Let U be an irreducible unitary representation of the compactquantum group G. There exists a unique positive, invertible element Q ∈ B(H) such that

(Q1/2 ⊗ 1)U(Q−1/2 ⊗ 1) is unitary and Tr(Q) = Tr(Q−1) .

In fact, the unitarity condition fixes Q up to a positive scalar. The normalization condition Tr(Q) =Tr(Q−1) then entirely determines Q.

Definition 4.6. The contragredient U c of an irreducible unitary representation U of G is definedas

U c := (Q1/2 ⊗ 1)U(Q−1/2 ⊗ 1) ,

where we have chosen Q as in Theorem 4.5.

Moreover, the positive real number Tr(Q) is called the quantum dimension of U and denoted bydimq(U).

Note that dimq(U) ≥ dim(U) because Tr(Q) = Tr(Q−1) and√

Tr(T ) Tr(T−1) ≥ n for any positiveinvertible matrix T ∈ Mn(C).

4.5 Representation theory and orthogonality relations

If U is a unitary representation of G on H, a vector ξ ∈ H is called an invariant vector if U(ξ⊗1) =ξ ⊗ 1. Denoting by ε the trivial representation of G, note that the subspace of invariant vectors ofU can be identified with Mor(U, ε).

Let U be an irreducible unitary representation of G and V any unitary representation. Since Vdecomposes as a direct sum of irreducibles, we can define the multiplicity of U in V in the obviousway and denote it as mult(U, V ). Note that mult(U, V ) equals the vector space dimension ofMor(U, V ).

Theorem 4.7 (Woronowicz). Let U be an irreducible unitary representation of G on H withcontragredient U c as in Definition 4.6. Take Q ∈ B(H) as in 4.5. Then, the following holds.

• The trivial representation appears with multiplicity 1 in U T© U c as well as in U cT© U .

• Choose an orthonormal basis (ei) for H. The (in fact base independent) formulas

t = Tr(Q)−1/2∑

i

ei ⊗Q1/2ei and s = Tr(Q)−1/2∑

i

Q−1/2ei ⊗ ei

define invariant unit vectors for U T© U c and U cT© U . It follows that

t∗(T ⊗ 1)t =Tr(QT )Tr(Q)

= (id⊗ h)(U(T ⊗ 1)U∗

),

s∗(1⊗ T )s =Tr(Q−1T )Tr(Q−1)

= (id⊗ h)(U∗(T ⊗ 1)U

),

(4.2)

for all T ∈ B(H).

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Proof. Define the linear bijection Θ : H ⊗H → B(H) : Θ(ξ ⊗ η) = ξη∗Q−1/2.

Exercise 7. The vector µ ∈ H ⊗H is invariant under U T©U c if and only if the matrix T := Θ(µ)satisfies U(T ⊗ 1)U∗ = T ⊗ 1.

By irreducibility of U , this last statement is equivalent with T being a multiple of the identitymatrix 1 ∈ B(H). So, the trivial representation has multiplicity 1 in U T©U c. Similarly for U c

T©U .

Exercise 8. Show that∑

i ei ⊗ ei is an invariant vector for U T© U . Deduce the invariance of t.

The only non-trivial thing still to prove are the formulas involving the Haar measure in (4.2). Theinvariance of s means that

U c13U23(s⊗ 1) = s⊗ 1 and so U23(t⊗ 1) = (U c)∗13(s⊗ 1) .

The second equation, combined with the unitarity of U c implies that

(s∗ ⊗ 1)(1⊗ U∗(T ⊗ 1)U

)(s⊗ 1) = s∗(1⊗ T )s⊗ 1 .

Applying the Haar state yields

s∗(1⊗ (id⊗ h)(U∗(T ⊗ 1)U)

)s = s∗(1⊗ T )s .

As in the proof of Theorem 4.4, the operator (id ⊗ h)(U∗(T ⊗ 1)U) belongs to End(U) and so, isscalar. It follows that this scalar is precisely s∗(1⊗T )s. The remaining formula with t is similar.

Formulas (4.2) yield as follows the orthogonality relations. Take irreducible unitary representationsU ∈ Mn(C(G)) and V ∈ Mm(C(G)). Since (id⊗ h)(U∗(T ⊗ 1)V ) and (id⊗ h)(U(T ⊗ 1)V ∗) belongto Mor(U, V ) it follows that

h(U∗ijVkl) = 0 = h(UijV∗kl) if U and V are inequivalent.

On the other hand, (4.2) says that

h(U∗ijUkl) =δjl (Q−1)ki

dimq(U)and h(UklU

∗ij) =

δkiQjl

dimq(U). (4.3)

Taking for V the trivial representation, we get h(Uij) = 0 for every irreducible representationinequivalent with the trivial representation. Since every unitary representation is a direct sum ofirreducibles, it follows that (id ⊗ h)(U) is the orthogonal projection on the subspace of invariantvectors for any unitary representation U .

As a result, we get the following. Let U, V be irreducible representations of G. Then the followingstatements are equivalent.

• V ∼= U c.

• V T© U contains the trivial representation.

• U T© V contains the trivial representation.

As a consequence, one gets the Frobenius reciprocity law :

mult(U, V T©W ) = mult(V c,W T© U c) = mult(W c, U cT© V )

for all irreducible unitary representations U, V,W .

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5 Algebra of matrix coefficients, universal and reduced C∗-algebras

Up to now, we told a lot of stories about unitary representations of a compact quantum group.But, does a compact quantum group necessarily admit more than just the trivial representation.The answer is of course yes and can be shown by constructing the regular unitary representation.

Applying the GNS-construction to the Haar state h on C(G) yields the Hilbert space separa-tion/completion L2(G) of C(G) together with the ∗-representation πred : C(G) → B(L2(G)). Weconsider C(G) as a dense subspace of L2(G).

We define now the right regular representation of G on L2(G). The easiest way to deal with this,and other, infinite dimensional representations on a Hilbert space H, is to identify M(K(H)⊗C(G))with the C∗-algebra of adjointable operators on the Hilbert C∗-module H ⊗ C(G).

Proposition 5.1 (Woronowicz). The formula

V (a⊗ 1) = ∆(a) for all a ∈ C(G)

defines a unitary representation V ∈ L(L2(G)⊗ C(G)) of G on L2(G).

If U ∈ L(H ⊗ C(G)) is a unitary representation of G on H, the elements

(ξ∗ ⊗ 1)U(η ⊗ 1) ∈ C(G)

are called matrix coefficients of U . Indeed, ifH is finite dimensional and if we choose an orthonormalbasis for H, we view U as a matrix over C(G) and the element (e∗i ⊗ 1)U(ej ⊗ 1) is exactly thematrix coefficient Uij .

Considering a, b ∈ C(G) as vectors in L2(G) on the left-hand side and as elements of C(G) on theright-hand side, we have

(a∗ ⊗ 1)V (b⊗ 1) = (h⊗ id)((a∗ ⊗ 1)∆(b)) .

It follows that the matrix coefficients of V span a dense subset of C(G).

One can then deduce the following result.

Theorem 5.2 (Woronowicz). Let G be a compact quantum group. Denote by Calg(G) the linearspan of all matrix coefficients of finite dimensional unitary representations of G. Then, Calg(G)is a dense ∗-subalgebra of C(G). Moreover, ∆ maps Calg(G) into the algebraic tensor productCalg(G)⊗alg Calg(G).

In fact, Calg(G) and ∆ form together a Hopf ∗-algebra. This can be used as a starting point tobuild a purely algebraic theory of compact quantum groups.

Definition 5.3. We define the following operator algebras associated with a compact quantumgroup G.

Cr(G) is the image of C(G) under πred,Cu(G) is the universal enveloping C∗-algebra of Calg(G),

L∞(G) is the weak closure of Cr(G) in B(L2(G)).

We have surjective ∗-homomorphisms Cu(G) → C(G) → Cr(G).

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To understand the previous definition, one has to look at the case G = Γ, defined in Example2.2. In that case, the irreducible representations of G are one-dimensional, labeled by the groupelements g ∈ Γ and given by the canonical unitaries ug in C∗(Γ). We then obtain the followingdictionary :

Cr(G) = C∗r (Γ) , Cu(G) = C∗(Γ) , L∞(G) = L(Γ) .

Taking into account the importance of the group C∗- and von Neumann algebras, this dictionarymotivates us to study Cr(G), Cu(G) and L∞(G) for concrete examples of compact quantum groups.

Recall that one of the equivalent definitions of amenability of a discrete group Γ, says that thecanonical homomorphism C∗(Γ) → C∗r (Γ) is an isomorphism. So, we give the following definition.

Definition 5.4. The compact quantum group G is called co-amenable if the canonical homomor-phism Cu(G) → Cr(G) is an isomorphism.

Observe that (4.3) can be used to show that the Haar state h is a KMS-state, both on Cr(G) andon Cu(G), with modular group given by

(id⊗ σht )(U) = (Qit ⊗ 1)U(Qit ⊗ 1) . (5.1)

In particular, the Haar state is a trace if and only if the Q-matrix of every irreducible representationis the identity matrix, i.e. U is unitary for every irreducible representation U .

Definition 5.5. A compact quantum group G is called of Kac type if one of the following equivalentconditions holds.

• The Haar state is a trace.

• For every irreducible representation U , U is unitary.

• The Q-matrix of every irreducible representation is the identity.

We finally benifit from Definition 5.3 to define morphisms and isomorphisms between compactquantum groups.

Definition 5.6. Let for i = 1, 2, Gi = (C(Gi),∆i) be compact quantum groups.

• We call π a morphism from G1 to G2 if π is a ∗-homomorphism

– either Calg(G2) → Calg(G1),

– or Cu(G2) → Cu(G2),

satisfying (π ⊗ π)∆1 = ∆2π .

• Replacing ∗-homomorphism by ∗-isomorphism, we get the definition of an isomorphism be-tween G1 and G2.

The important convention to notice, is the following : the same (i.e. isomorphic) compact quantumgroups can be defined by different underlying C∗-algebras. The reason for this lies in the followingexercise.

Exercise 9. Let Γ1,Γ2 be countable groups. Prove that Γ1 and Γ2 are isomorphic compact quantumgroups if and only if Γ1

∼= Γ2 as groups.

12

6 Examples of compact quantum groups

After all the abstract theory developed so far, it becomes urgent to introduce a few interestingexamples of compact quantum groups.

There are essentially two ways to obtain examples of compact quantum groups.

1. Start from a compact group G and deform the product on C(G) so that it becomes non-commutative. In this way, q-deformations of the classical compact groups have been defined.We will meet the easiest example of SUq(2) below.

2. Universal constructions. We start off immediately with an example of this kind.

6.1 The quantum groups Au(F )

Let G be a compact quantum group. Assume that C(G) is generated as a C∗-algebra by thecoefficients of the finite-dimensional unitary representation U . Note that this is not a very restrictivecondition : if G = Γ, it simply means that Γ is a finitely generated group. We know that U ∈Mn(C(G)) is a unitary matrix and that

∆(Uij) =∑

k

Uik ⊗ Ukj .

Moreover, since the Uij generate C(G) this last equation entirely determines ∆. But there is more.

Theorem 4.5 ensures the existence of a matrix F ∈ GLn(C) (take F = Q1/2) such that also FUF−1

is unitary.

Our first example is immediately the most universal compact quantum group, only imposing theconditions we have just found. See 6.2 for a precise statement.

Definition 6.1 (Wang & Van Daele). Let n ≥ 2 and take F ∈ GLn(C). Define the compactquantum group G as follows.

• C(G) is the universal unital C∗-algebra generated by the entries of the n × n matrix U ∈Mn(C(G)) subject to the relations

U and FUF−1 are unitaries.

• ∆ : C(G) → C(G)⊗ C(G) is uniquely defined such that U becomes a representation of G.

We denote G = Au(F ) and call it the universal unitary quantum group. We call U the fundamentalrepresentation of Au(F ).

Remark 6.2. Let G be any compact quantum group such that C(G) is generated by the matrixcoefficients of the n-dimensional unitary representation U . Theorem 4.5 implies the existenceof a matrix F ∈ GLn(C) and a surjective ∗-homomorphism π : C(Au(F )) → C(G) satisfying(π ⊗ π)∆ = ∆π.

The statement in the previous paragraph is the quantum version of the obvious statement thatevery n-generated countable group Γ is a quotient of the free group Fn.

13

Exercise 10. Let F1, F2 ∈ GLn(C) and suppose that there exists λ ∈ C and unitary matrices u, v ∈U(n) such that F2 = λuF1v. Prove that Au(F1) ∼= Au(F2). Also prove that Au(F ) ∼= Au(F−1).

It can be deduced from the representation theoretical results in Section 7 that these are the onlyisomorphisms between the quantum groups Au(F ). As a result, if Fi ∈ GLni(C) are normalizedsuch that Tr(F ∗i Fi) = Tr((F ∗i Fi)−1), we have Au(F1) ∼= Au(F2) if and only if n1 = n2 and F ∗1F1

has the same spectrum as F ∗2F2 or its inverse.

6.2 The quantum groups Ao(F )

In Subsection 4.1, we described the irreducible representations of the compact group SU(2). Itturns out that its fundamental representation is equivalent to its contragredient (in fact, all repre-sentations are like that and the fusion rules are commutative, given by (4.1)).

In the same spirit as with Au(F ), we take now the universal quantum group generated by a repre-sentation that is equivalent to its contragredient.

Definition 6.3 (Wang & Van Daele). Let n ≥ 2 and take F ∈ GLn(C). Define the compactquantum group G as follows.

• C(G) is the universal unital C∗-algebra generated by the entries of the n × n matrix U ∈Mn(C(G)) subject to the relations

U is unitary and U = FUF−1 .

• ∆ : C(G) → C(G)⊗ C(G) is uniquely defined such that U becomes a representation of G.

We denote G = Ao(F ) and call it the universal orthogonal quantum group. We call U the funda-mental representation of Ao(F ).

We note immediately that U = FUF−1, so that FF commutes with U . We want U to be an

irreducible representation, so that when dealing with Ao(F ), we impose FF to be a scalarmatrix and normalize F such that FF = ±1.

Take now a closer look to F =(

0 1−1 0

)and set G = Ao(F ). Then, the C∗-algebra C(G) is generated

by elements α, γ subject to the relation

U :=(α −γ∗γ α∗

)is a unitary representation.

Writing out the unitarity of U , we find that C(G) is commutative and we recognize that G = SU(2).

A more general 2-dimensional F is given by q ∈ [−1, 1] \ {0} and

Fq :=(

0 |q|1/2

− sgn(q)|q|−1/2 0

).

The resulting compact quantum group is Woronowicz’ celebrated SUq(2), with C(SUq(2)) generatedby elements α, γ subject to the relation

U :=(α −qγ∗γ α∗

)is a unitary representation.

14

Exercise 11. If F1, F2 ∈ GLn(C) satisfy FiFi = ±1, we set F1 ∼ F2 if there exists a unitaryu ∈ U(n) such that F2 = uF1u

t. Prove that for every F ∈ GL2(C) satisfying FF = ±1, there is aunique q ∈ [−1, 1] \ {0} satisfying F ∼ Fq. Prove that Ao(F1) ∼= Ao(F2) whenever F1 ∼ F2.

In fact, also the converse holds, but for this one needs the results of Section 7.

6.3 Digression

The operator algebras Cr(G) and L∞(G) are of the same interest as their much more studiedclassical counterparts C∗r (Γ) and L(Γ).

We discuss some known results and some open problems concerning the case G = Ao(F ) or Au(F ).First notice that if F = 1n is the n×n identity matrix, the orthogonality relations (4.3) imply thatthe Haar state on Cr(G) and L∞(G) is a trace.

1. In [4], Banica proved the simplicity of Cr(G) for G = Au(F ) and proved the uniqueness of thetrace when F = 1n. For arbitrary F , the Haar state is the unique KMS-state with modulargroup given by (5.1). The von Neumann algebras L∞(Au(F )) are factors, of type II1 if F isunitary and of type III if F is non-unitary.

2. The von Neumann algebras L∞(Au(1n)) are very closely related to the free group factorsL(Fm). In fact, Banica proved in [4] that L∞(Au(12)) ∼= L(F2), but his method cannot beused for bigger values of n. On the other hand, it is way beyond current techniques to refutethe isomorphism of L∞(Au(1n)) with some free group factor. It is also reasonable to expectthat L∞(Au(F )) for non-unitary F is isomorphic to a free Araki-Woods factor in the senseof Shlyakhtenko. Beyond 2× 2 matrices, we lack for the moments the methods to prove thisstatement.

3. Statement 1 is deduced from representation theoretic results for Au(F ) that we discuss inSection 7. In fact, the irreducible representations of Au(F ) form in some sense a free monoidand simplicity of Cr(G) can then be shown with the same methods as the simplicity of C∗r (Fm).On the other hand, if we shift to the case G = Ao(F ), things become more difficult. Now,the representation theory is the same as the one of the group SU(2) and is, in particular,commutative. Nevertheless, it is shown in [11] that Cr(Ao(1n)) is simple with unique tracewhenever n ≥ 3. It is also shown there that for n ≥ 3, L∞(Ao(1n)) is a solid II1 factor in thesense of Ozawa. When F is close enough to the n× n identity matrix and n ≥ 3, L∞(Ao(F ))is a type III factor. This is probably true for all non-unitary F , but this could not be provenin [11].

7 Representation theory for the examples above

The non-trivial task to determine all irreducible unitary representations and their fusion rules forthe compact quantum groups Ao(F ) and Au(F ) has been performed by Banica in [3, 4]. We startoff with the simplest case of Ao(F ) and give a sketch of proof.

15

7.1 Representation theory for Ao(F )

Theorem 7.1 (Banica [3]). Let F ∈ GLn(C) with FF = ±1. Let U be the fundamental represen-tation of G = Ao(F ). The irreducible representations of G are labeled by the natural numbers insuch a way that U0 is the trivial representation, U1 is the fundamental representation and the samefusion rules as for SU(2) hold :

Un T© Um∼= U|n−m| ⊕ U|n−m|+2 ⊕ · · · ⊕ Un+m . (7.1)

In particular, all Un are equivalent to their contragredient.

Fix F ∈ GLn(C) satisfying FF = c1 with c = ±1. Let G = Ao(F ) with fundamental representationU . Denote by U⊗k the k-fold tensor power U T© · · · T© U of U .

Denote by (ei) the standard basis of Cn and set

t := Tr(F ∗F )−1/2∑

i

ei ⊗ Fei .

Then, t is a unit invariant vector of U⊗2.

Lemma 7.2. Let k ≥ 2 and define orthogonal projections e1, ek−1 on (Cn)⊗k by the formula

es := 1⊗(s−1) ⊗ tt∗ ⊗ 1⊗(k−s−1) .

1. The orthogonal projections e1, . . . , ek−1 satisfy the Temperley-Lieb relations1

ei and ej commute if |i− j| ≥ 2 ,

ei = βeiejei if |i− j| = 1,

with β = Tr(F ∗F )−2.

2. The C∗-algebra End(U⊗k) is generated by e1, . . . , ek−1.

Moreover, the fundamental representation is irreducible.

Sketch of proof.

Exercise 12. Check that (t∗ ⊗ 1)(1 ⊗ t) = cTr(F ∗F )1 and deduce the Temperley-Lieb relations for

e1, . . . , ek−1.

We make the following claim : for all k, r ≥ 0, Mor(U⊗k, U⊗r) is linearly spanned by all meaningfulproducts of the operators 1⊗i ⊗ t ⊗ 1⊗j and their adjoints. Although this claim is not so easyto prove, it can be understood as follows. The definition of Ao(F ) only imposes the intertwinert. The only obvious operations that produce new intertwiners from old ones are tensoring withthe identity, multiplying, taking adjoints and taking linear combinations. In fact, doing so with t,one obtains what is called a concrete monoidal W ∗-category. Woronowicz’ reconstruction theoremassociates a compact quantum group to this concrete monoidal W ∗-category and it follows thatthere cannot be more intertwiners than the obvious ones.

1Recall from the course on subfactors by Dietmar Bisch that the sequence of Jones projections in the Jones towerof a subfactor satisfies this same Temperley-Lieb relations.

16

Exercise 13. Deduce the lemma from the claim above.

Lemma 7.2 says in particular that the ∗-algebra End(U⊗k) is a quotient of the Temperley-Liebalgebra. It will turn out that End(U⊗k) actually is the Temperley-Lieb algebra, but for the moment,we only keep in mind that dim End(U⊗k) is bounded by the dimension of the Temperley-Liebalgebra, which is known to be the k-th Catalan number Ck :

Ck =(2k)!

k! (k + 1)!.

In the course of the proof of Theorem 7.1, we make use of some representation theoretical results forthe group SU(2). Denote by Vk, k ∈ N its irreducible representations (see Subsection 4.1). Definepositive integers a(k, s) such that

V ⊗k1 =

k⊕s=0

a(k, s)Vs .

Note that a(k, k) = 1. Moreover, it is known that End(V ⊗k1 ) actually is the Temperley-Lieb algebra,

so that

Ck = dim End(V ⊗k1 ) =

k∑s=0

a(k, s)2 .

Proof of Theorem 7.1. We first prove by induction on k, the existence of inequivalent irreduciblerepresentations Uk of G = Ao(F ) such that U0 is the trivial representation, U1 = U is the funda-mental representation and

Uk T© U1∼= Uk−1 ⊕ Uk+1 for all k ≥ 1 . (7.2)

Suppose that for s ≥ 1, we have constructed U1, . . . , Us satisfying (7.2) for all 1 ≤ k ≤ s− 1. Then,Us is contained once in Us−1 T©U1. By Frobenius reciprocity and because U1

∼= U c1 , we get that Us−1

is contained once in Us T© U1 and we define Us+1 as its orthogonal complement. By construction,

U⊗(s+1)1

∼=s+1⊕k=0

a(s+ 1, k)Uk .

Hence,

dim End(U⊗(s+1)1 ) ≥

s+1∑k=0

a(s+ 1, k)2 (7.3)

with equality holding if and only if Us+1 is irreducible and inequivalent to U0, . . . , Us. But theleft-hand side of (7.3) is smaller or equal to Cs+1 by Lemma 7.2, while the right-hand side of(7.3) equals Cs+1 by the observations preceding this proof. So, we get equality and the Uk areconstructed.

Now (7.2) implies that the fusion rules (7.1) between the representations Uk hold. So, the trivialrepresentation is contained in Uk T© Uk, implying that Uk

∼= U ck . Finally, it follows that the matrix

coefficients of the representations Uk span the ∗-algebra generated by the matrix coefficients of thefundamental representation. So, we have found all irreducible representations.

Exercise 14. Prove the following nice converse of Theorem 7.1. If G has the same fusion rules asSU(2), there exists a matrix F ∈ GLn(C) such that FF = ±1 and G ∼= Ao(F ). Hint : the invariantvector of U1 T© U1 defines F .

17

7.2 Digression : about the co-amenability of the Ao(F )

We prove the following result.

Proposition 7.3 (Banica, [3]). Let F ∈ GLn(C) and FF = ±1. Then, Ao(F ) is co-amenable inthe sense of 5.4 if and only if n = 2.

We first introduce a tool that is interesting in its own right : the character of a unitary represen-tation. If U is a finite-dimensional unitary representation of G on H, we define

χ(U) = (Tr⊗id)(U)

and call χ(U) the character of U .

Exercise 15. Check that

χ(U ⊕ V ) = χ(U) + χ(V ) , χ(U T© V ) = χ(U)χ(V ) , χ(U c) = χ(U)∗ .

We have already observed that (id⊗h)(U) is the orthogonal projection onto the U -invariant vectors.As a result,

h(χ(U)) = mult(ε, U) .

Combining this observation with Exercise 15 and Frobenius reciprocity, we get for all finite-dimensional unitary representations U and V ,

dim Mor(U, V ) = h(χ(U)∗χ(V )) .

We illustrate all this with the example of Ao(F ). Let U be the fundamental representation of Ao(F ).Since U ∼= U c, it follows that χ(U) is a self-adjoint element of C(Ao(F )) and that h(χ(U)n) =mult(ε, U⊗n). This last multiplicity is entirely determined by the fusion rules and hence the sameas for SU(2). In the SU(2) case, it is known that

mult(ε, U⊗n) =

{Ck if n = 2k,0 if n is odd.

We conclude that

h(χ(U)n) =

{Ck if n = 2k,0 if n is odd.

=12π

∫ 2

−2xn

√4− x2 dx .

In words, this reads as follows : χ(U) is distributed with respect to the Haar state as a semi-circularvariable of radius 2.

The relevance of the distribution of χ(U) for amenability issues, lies in the following lemma.

Lemma 7.4 (Skandalis, Banica [4]). Let G be a compact quantum group with unitary representationU of dimension n < ∞. Suppose that the matrix coefficients of U generate C(G) as a C∗-algebra.Then, the following conditions are equivalent.

1. G is co-amenable.

18

2. The support of the spectral measure of Re(χ(U)) with respect to the Haar state h, contains n.

3. The spectrum of Re(πred(χ(U))) contains n.

Proof (rather sketchy). Conditions 2 and 3 are equivalent by spectral theory and the faithfulnessof h on L∞(G).

Suppose condition 1 holds. So, πred is a ∗-isomorphism that we de not write anymore. Since themap

ε : Calg(G) → C : (id⊗ ε)(V ) = 1 for every irreducible representation V ,

is easily seen to be a ∗-homomorphism (called the co-unit), we obtain the ∗-homomorphism ε :Cr(G) → C satisfying (id⊗ ε)(U) = 1. It follows that ε(n− Re(χ(U))) = 0, so that n− Re(χ(U))is non-invertible and hence, n belongs to the spectrum of Re(χ(U)).

Suppose finally that condition 3 holds. Since 2 Re(χ(U)) = χ(U⊕U c), we may replace U by U⊕U c

and assume that U ∼= U c. So, n belongs to the spectrum of πred(χ(U)) and we can take a sequenceof unit vectors ξk in L2(G) such that

‖πred(χ(U))ξk − dim(U)ξk‖ → 0 . (7.4)

Let ω be a weak∗ accumulation point in the state space of Cr(G) of the sequence of vector states〈ξk, · ξk〉. We claim that ω ◦ πred = ε. So, we have to prove that (id ⊗ ωπred)(V ) = 1 for everyirreducible unitary representation V . Since V is contained in some tensor power of U and because(7.4) remains true for tensor powers of U by multiplying, we may assume that V is contained in U .

By (7.4), 〈ξk, πred(χ(U))ξk〉 → dim(U) and so ω(πred(χ(U))) = dim(U). But then (id⊗ ωπred)(U)is a matrix of norm at most 1 and trace dim(U). That is only possible if (id⊗ωπred)(U) = 1. Thenalso (id ⊗ ωπred)(V ) = 1. Hence, our claim is proved. In particular, ω is a ∗-homomorphism fromCr(G) to C.

So, we have obtained a character ω on Cr(G) satisfying ω ◦ πred = ε. For people acquaintedwith amenability of discrete groups, this statement reads as ‘the trivial representation is weaklycontained in the regular representation’, which is equivalent with amenability. This is still true asthe following argument shows. The invariance of the Haar measure allows for any compact quantumgroup to define uniquely the unital ∗-homomorphism

α : Cr(G) → Cu(G)⊗ Cr(G) : α ◦ πred = (id⊗ πred)∆ .

But then, (id⊗ω)α is a ∗-homomorphism from Cr(G) to Cu(G) that is easily seen to be the inverseof πred. Hence, G is co-amenable.

Proof of Proposition 7.3. Let U be the fundamental representation of Ao(F ) and χ(U) its character.We have seen above that χ(U) is distributed w.r.t. h as a semi-circular variable with radius 2. So,the support of the spectral measure of χ(U) w.r.t. h equals [−2, 2]. By 7.4, Ao(F ) is co-amenableif and only if dim(U) = 2.

7.3 Representation theory for the Au(F )

The representation theory of Au(F ) is considerably more difficult to obtain. We only state theresult and refer to [4] for a proof.

We make use of the free monoid N ∗ N generated by α, β and equipped with the unique anti-multiplicative involution x 7→ x satisfying α = β.

19

Theorem 7.5 (Banica [4]). Let F ∈ GLn(C). The irreducible representations of Au(F ) are labeledby the free monoid N ∗ N with generators α, β in such a way that

• Uα is the fundamental representation,

• UxT© Uy =

⊕z with x=x0z,y=zy0

Ux0y0,

• the contragredient of Ux is Ux.

Remark 7.6. The Au(F ) are very far from being co-amenable. As we observed above, Cr(Au(F ))is even a simple C∗-algebra and so certainly does not have a one-dimensional representation ε :Cr(Au(F )) → C.

The same can be seen using Lemma 7.4. It can be shown that the character χ(U) of the fundamentalrepresentation U of Au(F ) is ∗-distributed w.r.t. h as a circular variable of radius 2 in the sense ofVoiculescu. So, still w.r.t. h, the spectral radius of Re(χ(U)) is

√2, which is always strictly smaller

than dim(U).

8 The dual of a compact quantum group, discrete quantum groups

We now construct the dual of a compact quantum group G, which will be the discrete quantumgroup G.

We are of course guided by the case G = Γ where we should obtain in some way G = Γ. Wealready noticed that in that case, Γ labels exactly the irreducible representations of G, which areone-dimensional and given by the canonical unitaries ug ∈ C(G) = C∗(Γ). This motivates thefollowing definition.

Definition 8.1. Let G be a compact quantum group. Let Irred(G) be the set of equivalence classesof irreducible unitary representations of G and choose representatives, i.e. unitary representationsUx on Hx for all x ∈ Irred(G). Define

c0(G) :=⊕

x∈Irred(G)

B(Hx) ,

`∞(G) :=∏

x∈Irred(G)

B(Hx) .

We also want a group like structure on `∞(G). In fact, there exists a normal ∗-homomorphism

∆ : `∞(G) → `∞(G)⊗`∞(G)

satisfying ∆(a)T = Ta whenever T ∈ Mor(Ux, UyT© U z) , a ∈ `∞(G) .

Some clarifying comments : here and it what follows, ⊗ denotes the tensor product of von Neumannalgebras. Further, if T ∈ Mor(Ux, Uy

T© U z), we interpret the equation ∆(a)T = Ta as follows :(∆(a)(py ⊗ pz)

)T = T (apx) .

Here, and again in what follows, we denote by (px)x∈Irred(G) the minimal central projections of`∞(G).

20

The duality between G and G is expressed by the following unitary operator :

V :=⊕

x∈Irred(G)

Ux .

Note that V ∈ M(c0(G)⊗ C(G)) and that (id⊗∆)(V) = V12 V13.

Exercise 16. Prove that(∆⊗ id)(V) = V13 V23 (8.1)

and deduce the co-associativity (∆ ⊗ id)∆ = (id ⊗ ∆)∆. Hint. It is sufficient to check (8.1)after multiplying the left-hand side and the right-hand side of (8.1) with T ⊗ 1 for an arbitraryT ∈ Mor(Ux, Uy

T© U z).

Exercise 17. Let Γ be a countable group and set G = Γ. Identify `∞(G) with `∞(Γ) in such away that

∆ : `∞(Γ) → `∞(Γ)⊗`∞(Γ) = `∞(Γ× Γ) : (∆(a))(x, y) = a(xy) for all a ∈ `∞(Γ) , x, y ∈ Γ .

Remark 8.2. Van Daele has given an intrinsic definition of a discrete quantum group and thedual of such a discrete quantum group is a compact quantum group. A more general, self-dualframework that we do not discuss in this course, is the one of locally compact quantum groups.

9 Actions of compact quantum groups and subfactors

There are basically two relations between groups and operator algebras : as above, groups directlygenerate operator algebras, but on the other hand groups act on C∗- and von Neumann algebras byautomorphisms. The same is true for quantum groups and yields intimate relations with subfactortheory.

Definition 9.1. An action of a compact quantum group G on a von Neumann algebra M is aunital faithful normal ∗-homomorphism α : M →M⊗L∞(G) satisfying

(α⊗ id)α = (id⊗∆)α .

Given an action α of G on M , one defines and denotes the fixed point algebra by

Mα := {a ∈M | α(a) = a⊗ 1} .

Exercise 18. Let α be an action of G on M and h the Haar state of G.

1. Prove that E := (id⊗h)α is a conditional expectation of M onto the fixed point algebra Mα.

2. A normal state ω on M is called invariant if (ω ⊗ id)α(a) = ω(a)1 for all a ∈M . Prove thatω ◦E is invariant for any normal state ω on Mα and that every normal invariant state ariseslike this.

In a similar way, actions of compact quantum groups on unital C∗-algebras are defined.

Definition 9.2. An action of a compact quantum group G on a unital C∗-algebra B is a unital∗-homomorphism α : B → B ⊗ C(G) satisfying

(α⊗ id)α = (id⊗∆)α and α(B)(1⊗ C(G)) is total in B ⊗ C(G).

To understand the subtle nuance between Def. 9.1 (supposing α is faithful) and Def. 9.2 (supposingthe density condition), we refer to Remark 9.9.

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9.1 Minimal actions and subfactors

As a motivation to study actions of compact quantum groups, we immediately present the con-struction of the associated Wassermann subfactors. So, take an action α of a compact quantumgroup G on a factor M . Choose a finite-dimensional representation U ∈ B(H)⊗ C(G) of G on H.We also view U inside B(H)⊗ L∞(G) using πred.

Exercise 19. Prove that

β : B(H)⊗M → B(H)⊗M⊗L∞(G) : β(a) = U13(id⊗ α)(a)U∗13

defines an action of G on B(H)⊗M .

We are now interested in the inclusion 1 ⊗Mα ⊂ (B(H)⊗M)β. For the moment, this is just aninclusion of von Neumann algebras. In order to say more, we introduce a special kind of actions.

Definition 9.3. An action α of a compact quantum group G on a factor M is called minimal if

• the relative commutant of Mα inside M is trivial, i.e. (Mα)′ ∩M = C1,

• a faithfulness condition holds : the irreducible representations of G contained in α (in thesense of 9.8) generate2 Irred(G).

For the moment, we do not bother about the faithfulness condition. It excludes for instance thatthe trivial action on an arbitrary factor α : M → M⊗L∞(G) : α(a) = a ⊗ 1 would be calledminimal. (The only irreducible representation contained in the trivial action, is of course the trivialrepresentation.)

Suppose now that α is a minimal action of G on a factor M . Take an irreducible finite-dimensionalunitary representation U of G on H and consider the above constructed Wassermann inclusion1 ⊗Mα ⊂ (B(H)⊗M)β. We claim that this inclusion is in fact an irreducible subfactor, meaningthat the relative commutant of 1⊗Mα inside (B(H)⊗M)β is trivial. Indeed, first of all

(1⊗Mα)′ ∩ (B(H)⊗M)β ⊂ (1⊗Mα)′ ∩ (B(H)⊗M) = B(H)⊗1 .

But a ⊗ 1 belongs to (B(H)⊗M)β if and only if U(a ⊗ 1)U∗ = a ⊗ 1. So, the irreducibility of Uyields the claim.

If now moreoverM is a II1 factor3, the Wassermann subfactor is an irreducible inclusion of II1 factorsof index dim(U)2. The Jones tower of the Wassermann subfactor and the corresponding lattice ofrelative commutants can be explicitly determined as follows. Define for any finite-dimensionalunitary representation U , the factor

N(U) := (B(H)⊗M)β where β : B(H)⊗M → B(H)⊗M⊗L∞(G) : β(a) = U13(id⊗ α)(a)U∗13 .

When U and V are finite-dimensional representations, we have natural embeddings

N(U) ↪→ N(V T© U) : a 7→ 1⊗ a .

2We say that a subset F ⊂ Irred(G) generates Irred(G) if every irreducible representation of G is contained in atensor product of elements of F .

3This imposes G to have a tracial Haar state, see [10] and the concluding remarks at the end of Subsection 9.2

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The Jones tower of the Wassermann subfactor can then be identified with the tower

Mα = N(ε) ↪→ N(U) ↪→ N(U cT© U) ↪→ N(U T© U c

T© U) ↪→ · · · (9.1)

Minimality of α implies that the first relative commutants are given by

C ↪→ C = End(U) ↪→ End(U cT© U) ↪→ End(U T© U c

T© U) ↪→

So, to a certain extent, the representation theory of G is encoded in the Wassermann subfactor.

In fact, it is not necessary to assume that M is a II1 factor. We briefly explain this. So, let M beany factor with a minimal action α of a compact quantum group G.

Exercise 20. Let U be an irreducible representation of G on H. Define the state ϕU on B(H) bythe formula

ϕU (a) =Tr(Q−1a)

Tr(Q),

where Q is given by Theorem 4.5. Use (4.2) to prove that

(ϕU ⊗ id)(U∗(a⊗ 1)U) = ϕU (a)1 for all a ∈ B(H) .

Deduce that ϕU ⊗ id restricts to a conditional expectation of (B(H)⊗M)β onto Mα.

So, the Wassermann subfactor (1 ⊗Mα) ⊂ (B(H) ⊗M)β comes with a canonical conditionalexpectation. There is a theory of Jones index and a construction of the Jones tower for inclusionsof arbitrary factors with conditional expectations. The Jones tower of the Wassermann subfactor isstill given by (9.1) and the index of the conditional expectation constructed in Exercise 20 is givenby the square of the quantum dimension dimq(U)2.

All of the above only becomes really interesting once we have examples of minimal actions ofcompact quantum groups. This is not a trivial issue, as illustrated by the following remark. Moredetails follow in the next subsection.

Remark 9.4. Suppose that the compact quantum group SUq(2) admits a minimal action on aninjective factor. The Wassermann subfactor associated with the fundamental representation is thenan irreducible injective subfactor of index |q+1/q|2. Sorin Popa has shown (unpublished) that notall values greater than 4 can arise as the index of an irreducible injective subfactor. This resultimplies that not all SUq(2) admit a minimal action on an injective factor.

Nevertheless, we will see below that all compact quantum groups admit minimal actions on certainnon-injective factors. Combined with the Wassermann subfactor construction, this has been usedby Shlyakhtenko and Ueda to construct irreducible subfactors of L(F∞) of arbitrary index.

9.2 Constructions of minimal actions

Before constructing our first examples of minimal actions, we have to construct the factors on whichwe will act.

23

Free product von Neumann algebras

We define the free product of two von Neumann algebras M1,M2 equipped with faithful normalstates ω1, ω2. We get

(M,ω) = (M1, ω1) ∗ (M2, ω2)

in such a way that M contains state preserving copies of M1 and M2 and such that Voiculescu’sfreeness condition holds :

ω(a0a1 · · · an) = 0 whenever a0, a1, . . . , an are alternatingly in M1 and M2

and satisfy ω(ai) = 0 for all i.

The following is the least deep phrase of these lecture notes : in a free product, there is almost nocommutativity and the free product is almost always a factor.

Exercise 21. Let Γ1,Γ2 be countable groups and consider the group von Neumann algebrasL(Γ1),L(Γ2) with their natural tracial states τ1, τ2. Identify L(Γ1 ∗ Γ2) = (L(Γ1), τ1) ∗ (L(Γ2), τ2).

If Γ1 = Γ2 = Z/2Z, the free product Γ1 ∗ Γ2 is the infinite dihedral group. This group is not anICC group and so, the free product of C2 with itself (w.r.t. the equally distributed traces) is not afactor. Basically, that is the only case where a free product fails to be a factor.

For our purposes, the following little lemma is sufficient.

Lemma 9.5. Let (M1, ω1) be a von Neumann algebra with a faithful normal state. Let (M2, ω2) bea II1 factor with its tracial state ω2. Then, the relative commutant of M2 inside M is trivial.

Sketch of proof. Suppose that a ∈ M ′2 ∩ M . We want to show that a ∈ C1. Because M2 is a

factor, it is sufficient to prove that a ∈ M2. Denote by E2 : M → M2 the unique state preservingconditional expectation. Replacing a be a−E2(a), we may assume that E2(a) = 0 and we have toprove that actually a = 0.

Take a sequence of unitaries un in M2 tending to 0 weakly. We claim that ω(bundu∗n) → 0 whenever

b, d ∈ M with E2(b) = E2(d) = 0. Although it takes some technical skills to justify this, it issufficient to prove the claim if moreover b and d are alternating products of elements in M1 andM2. In that more special case, the claim follows from freeness and the fact that un tends to 0weakly.

Using the claim, it follows that ω(a∗unaun) tends to 0. But, ω(a∗unaun) = ω(a∗a). So, ω(a∗a) = 0and hence, a = 0.

Free product actions (Ueda)

Assume now that α1, α2 are state preserving actions of a compact quantum group G on (M1, ω1),(M2, ω2). There is then a unique action α of G on (M,ω) = (M1, ω1) ∗ (M2, ω2) extending both α1

and α2. Although this is not very hard, we did not introduce enough free product theory to give ashort proof.

Combined with Lemma 9.5, we obtain the following theorem.

Theorem 9.6. Let G be a compact quantum group and α a faithful state preserving action of Gon (M,ω). Take for instance the action of G on itself given by the comultiplication ∆ : L∞(G) →L∞(G)⊗L∞(G).

Then, the free product of the action α on (M,ω) and the trivial action on an arbitrary II1 factor,is minimal. In particular, every compact quantum group admits minimal actions on factors.

24

Concluding remarks

Inspired by Connes’ T -invariant for factors, I introduced in [10] an invariant T (G) for any compact4

quantum group G. As for the T -invariant of a factor, T (G) is a subgroup of the group of positivereal numbers. It is shown that whenever G admits a minimal action on the factor M , the inclusionT (M) ⊂ T (G) holds. Therefore, if T (G) 6= R, the quantum group G cannot act minimally on a II1factor.

For more results on minimal actions of quantum groups and constructions, see [7, 10] and referencestherein. Let me for instance highlight the following result of [10] : there exists a type III1 factor(a concrete free Araki-Woods factor) on which every (second countable) compact quantum groupadmits a minimal action.

9.3 Quantum automorphism groups

We defined actions of compact quantum groups on von Neumann algebras and noticed that theyalways admit invariant states.

On the other hand, compact groups can often be defined by their action on some structure preservingsomething that yields the compactness. The same is true for compact quantum groups and wepresent the simplest example now. More elaborate examples arise in non-commutative geometryas the isometry quantum groups of some of Connes’ spectral triples. Other examples include thequantum groups acting on finite graphs, introduced and studied by Banica, Bichon and others.

Definition 9.7. Let (B,ω) be a finite-dimensional von Neumann algebra with faithful state ω.The compact quantum group G = Aaut(B,ω) is defined as the universal compact quantum groupacting on B and preserving ω.

An abstract characterization of the unital C∗-algebra C(G) can be given as follows.

• There exists a unital ∗-homomorphism α : B → B⊗C(G) satisfying (ω⊗ id)α(a) = ω(a)1 forall a ∈ B.

• {(ρ⊗ id)α(a) | ρ ∈ B∗, a ∈ B} generates C(G) as a C∗-algebra.

• Whenever A is a unital C∗-algebra and β : B → B ⊗ A a unital ∗-homomorphism satisfying(ω ⊗ id)β(a) = ω(a)1 for all a ∈ B, there exists a unital ∗-homomorphism π : C(G) → Asatisfying β = (id⊗ π)α.

The comultiplication ∆ on C(G) is uniquely characterized by the action formula

(id⊗∆)α = (α⊗ id)α .

In order to prove correctly the existence of a unital C∗-algebra C(G) with the above universalproperties, one first chooses a basis ai of B with dual basis ρi of B∗. Consider formally Uij =(ρi⊗ id)α(aj). Writing out that α is a unital ∗-homomorphism preserving ω, yields relations on theUij . Then, C(G) is defined as the universal C∗-algebra with generators Uij and these relations.

4In fact, the T -invariant is defined for any locally compact quantum group.

25

9.4 Spectral theory of compact quantum group actions

Fix a compact quantum group G and an action α of G on a unital C∗-algebra B.

Definition 9.8. If U is an irreducible representation of G on H, we say that α contains U if Bcontains an α-invariant vector subspace on which α acts in a way equivalent with U . This can beexpressed by the following two equivalent statements.

• There exists a non-zero vector ξ ∈ H ⊗B satisfying (id⊗ α)(ξ) = ξ12U13 .

• Choose an orthonormal basis for H and put n = dimU . There exists a non-zero n-tuple(a1, . . . , an) ∈ Bn satisfying

α(ai) =∑

j

aj ⊗ Uji for all 1 ≤ i ≤ n . (9.2)

Exercise 22. Define BU as the linear span of all entries of all dim(U)-tuples satisfying (9.2). DefineB as the linear span of all the BU . Prove that B is a ∗-subalgebra of B. We call B the spectralsubalgebra of B.

Let U be an n-dimensional irreducible representation of G with matrix coefficients Uij . Fix a ∈ Band fix 1 ≤ k ≤ n. Define the n-tuple

ai = (id⊗ h)(α(a)(1⊗ U∗ik)) .

Prove that (a1, . . . , an) satisfies (9.2). Deduce that B is a dense ∗-subalgebra of B. Hint. To provethe density, use the fact that α(B)(1⊗ C(G)) is total in B ⊗ C(G).

Remark 9.9. The spectral subalgebra can be defined in a similar way for actions on von Neumannalgebras. It is a strongly∗ dense subalgebra, but for the following reason, this is more difficult toprove.

In the definition of an action on a von Neumann algebra, α(M)(1⊗ L∞(G)) is not assumed to be(weakly) total in M⊗L∞(G). The reason for this is that the faithfulness of α allows to prove thisdensity condition, but that is rather non-trivial.

10 Infinite tensor product actions and Poisson boundaries

The minimal actions constructed in Theorem 9.6 are on free product factors and do not yieldminimal actions on injective factors. This is not surprising. It is not too hard to show that if Gadmits a minimal action on an injective factor, then G is an amenable discrete quantum group (see[10]). Moreover, Tomatsu showed that amenability of G is equivalent with co-amenability of G, inthe sense of Definition 5.4.

So, we can only expect to find minimal actions on injective factors for co-amenable compact quan-tum groups. But even then, life is not easy, as suggested by Remark 9.4. In fact, and this is quiteremarkable, the quest for minimal actions on injective factors will lead us to consider random walkson discrete quantum groups and to the computation of their Poisson boundaries. And that will bethe main part of the rest of this lecture series.

26

Infinite tensor product von Neumann algebras

If (Mn, ωn) is a sequence of von Neumann algebras Mn equipped with faithful normal states ωn,one constructs as follows the infinite tensor product von Neumann algebra with its infinite productstate. First of all, we have the obvious direct limit ∗-algebra

alg-⊗

n

(Mn, ωn)

equipped with its natural state ω. Next, use ω to perform the GNS-construction and define Mas the weak closure of the algebraic infinite tensor product acting on the GNS Hilbert space. Byconstruction, we have the following crucial properties.

1. ω is a faithful normal state on M .

2. There are ω-preserving conditional expectations

En : M →⊗n

k=0Mk : En(a0 ⊗ a1 ⊗ · · · ⊗ ar ⊗ 1⊗ 1⊗ · · · ) =

( r∏i=n+1

ωi(ai))a0 ⊗ · · · ⊗ an .

3. For all a ∈M , the bounded sequence En(a) converges ∗-strongly to a.

We write (M,ω) =⊗

n

(Mn, ωn) .

Denote Mn =⊗n

k=0Mk. A bounded sequence of elements an ∈ Mn is called a martingale ifEn(am) = an for all m ≥ n. If a ∈M , the sequence (En(a)) is obviously a martingale, but also theconverse holds. Indeed, if (an) is a martingale, set m = supn ‖an‖ and choose a weak∗-accumulationpoint a ∈M of the sequence (an) in the ball with radius m. By continuity of Ek, we get that Ek(a)is a weak∗-accumulation point of the sequence (Ek(an))n. This sequence is constantly ak for n ≥ k.So, Ek(a) = ak and it follows that ak → a ∗-strongly.

Infinite tensor product actions of compact groups

Let (M,ω) be a factor with faithful normal state ω. Suppose that the compact group G acts on Mby automorphisms αg preserving ω. We consider

(M, ω) :=⊗∞

k=0(M,ω)

with diagonal action of G on M defined by

βg(a0 ⊗ · · · ⊗ an ⊗ 1⊗ 1⊗ · · · ) = αg(a0)⊗ · · · ⊗ αg(an)⊗ 1⊗ 1⊗ · · · .

Theorem 10.1. If the action (αg) is faithful (meaning that αg 6= id if g 6= e), the action (βg) isminimal.

Proof. Since (αg) is faithful, also (βg) is faithful. Suppose that a ∈ (Mβ)′ ∩M. Define for everyn ∈ N the automorphism

σn ∈ Aut(M) : σn(a0 ⊗ · · · ⊗ an ⊗ an+1 ⊗ · · · ⊗ 1⊗ 1⊗ · · · ) = an ⊗ a0 ⊗ · · · ⊗ an−1 ⊗ an+1 ⊗ · · ·

27

Check that σn commutes with βg for all g. Hence, σn(a) ∈ (Mβ)′ ∩M for all n.

Next define the embedding ψ : M → M : ψ(a0 ⊗ a1 ⊗ · · · ) = 1 ⊗ a0 ⊗ a1 ⊗ · · · . We claim that‖σn(b) − ψ(b)‖2 → 0 for all b ∈ M, where ‖d‖2 =

√ω(d∗d). In fact, it is sufficient to check the

claim when b is of the form a0 ⊗ · · · ⊗ ak ⊗ 1⊗ 1⊗ · · · and in this case, the claim is obvious. As aresult, ψ(a) ∈ (Mβ)′ ∩M.

Iterating ψ, it follows that 1⊗n ⊗ a belongs to (Mβ)′ ∩M for all n. Here and in the rest of theproof, we make the identification M = M⊗ · · ·⊗M⊗M. Choose b1, . . . , bn and b in M . Integratingwith respect to the Haar measure, we get that∫

G(αg(b1)⊗ · · · ⊗ αg(bn)⊗ αg(b)) dg ⊗ 1⊗ 1⊗ · · ·

belongs to Mβ and hence commutes with 1⊗n⊗a. Choose ρ1, . . . , ρn ∈M∗ and define the functionF ∈ C(G) as

F (g) = ρ1(αg(b1)) · · · ρn(αg(bn)) .

It follows that a commutes with (∫GF (g)αg(b) dg

)⊗ 1⊗ 1⊗ · · ·

for all b ∈ M and all F ∈ C(G) of the form above. The faithfulness of the action implies thatthe functions F span a norm dense subspace of C(G). Approximating the Dirac function in e, weconclude that a commutes with b⊗ 1⊗ 1⊗ · · · for all b ∈M .

Since M is a factor, it follows that a = 1 ⊗ a′. But then, a′ ∈ (Mβ)′ ∩M. Repeating the sameargument, a′ = 1⊗ a′′. Continuing in this way, it follows that a ∈ C1.

Corollary 10.2. Every second countable compact group admits a minimal action on the hyperfiniteII1 factor.

There are several other constructions of minimal actions of compact groups. We state the followingresult, associating a minimal action to any countable group Γ. The proposition is not very hard toprove, but one needs some acquaintance with von Neumann algebra crossed products in order todo so.

Proposition 10.3. Let (M, τ) be a von Neumann algebra with faithful tracial state. Note that Mis not assumed to be a factor. Let (αg) be a faithful τ preserving action of a compact group G onM . Let Γ be any countably infinite group. Define the action (βg) of G on the Bernoulli crossedproduct (⊗

Γ

(M, τ))

o Γ

such that (βg) acts trivially on Γ and is the infinite tensor product action on⊗Γ

(M, τ).

Then, (βg) is a minimal action.

In particular, we could start from the action of G on itself, i.e. G acting on L∞(G) preserving theHaar state and get a minimal action of G on

L∞(GΓ

)o Γ (10.1)

28

for any countably infinite group Γ.

Two actions (αg) and (βg) on the von Neumann algebras M and N are called conjugate if thereexists a ∗-isomorphism π : M → N satisfying βg = π ◦ αg ◦ π−1. An amazing accumulation ofmathematics yields the following theorem, very far beyond the scope of these lectures.

Theorem 10.4 (Ocneanu, Popa, Wassermann). All minimal actions of a compact group G on thehyperfinite II1 factor are conjugate.

One should oppose Theorem 10.4 to the many constructions of minimal actions described above.Indeed, Connes’ celebrated uniqueness theorem for injective II1 factors says that the II1 factorsin Proposition 10.3 are all isomorphic to the hyperfinite II1 factor, whenever M is injective andΓ is amenable. In particular, whenever Γ is amenable, the II1 factor in (10.1) is hyperfinite andall these minimal G-actions are conjugate. It is by no means possible to write down explicitly theconjugation though.

We finally note that Theorem 10.4 has recently been extended by Masuda and Tomatsu to allminimal actions on the hyperfinite II1 factor of co-amenable compact quantum groups of Kac type.And a fortiori, the theorem of Masuda and Tomatsu is also beyond the scope of these lectures.

Infinite tensor product actions of compact quantum groups

In order to obtain minimal actions of quantum groups on injective factors, Theorem 10.1 suggeststo construct infinite tensor product actions. But, given an action α of G on M , there is no naturaldiagonal action of G on M⊗M . It is possible though to define a kind of diagonal action for inneractions. That is what we do now.

Fix a unitary representation of a compact quantum group G on H. Define the inner action

α(1) : B(H) → B(H)⊗L∞(G) : α(1)(a) = U(a⊗ 1)U∗ .

Choose a faithful normal state ω on B(H) invariant under α(1).

Exercise 23. Perform the following steps, leading to the infinite tensor product action α(∞) of Gon M =

⊗∞k=0(B(H), ω).

1. Denote by U⊗n the n-fold tensor product of U . The formula α(n) : B(H⊗n) → B(H⊗n)⊗L∞(G) :α(n)(a) = U⊗n(a⊗ 1)(U⊗n)∗ defines an action of G on B(H⊗n) preserving ω⊗n.

2. Denoting in : B(H⊗n) → B(H⊗(n+1)) : in(a) = a⊗1, we have α(n+1)(in(a)) = (in⊗id)α(n)(a).

3. Construct α(∞).

4. Denote by En : M→ B(H⊗n) the canonical conditional expectation and check that

α(n) ◦ En = (En ⊗ id)α(∞) .

We now address the following two issues. We denote α = α(∞).

• Faithfulness of α.

• Computation of the relative commutant (Mα)′ ∩M.

29

Note that α is faithful if and only if every irreducible representation of G appears in some U⊗nT©

(U c)⊗n. But, in order to avoid some technicalities in the computation of (Mα)′ ∩M, we are moresevere and make the following assumptions.

Assumption. U contains the trivial representation and generates Irred(G) in the sense that everyirreducible representation of G appears in some tensor power of U .

The formula V = (π⊗id)(V) establishes a bijective correspondence between unitary representationsV of G and representations π of the von Neumann algebra `∞(G). So, take π such that U =(π⊗ id)(V). Set ψ = ω ◦ π. Then, ψ is a normal state on `∞(G) and we can define the convolutionoperator

P : `∞(G) → `∞(G) : P (a) = (id⊗ ψ)∆(a) . (10.2)

Later, it will become clear that ψ is necessarily a very special type of state and that P should beinterpreted as the Markov operator associated to a random walk on the discrete quantum group G.

For the moment, we only retain that P is a completely positive unital map and we are interestedin the P -harmonic elements

H∞(G, P ) := {a ∈ `∞(G) | P (a) = a} .

Theorem 10.5 (Izumi [7]). Let G be a compact quantum group and U a unitary representationof G. Assume that U is generating and contains the trivial representation. Construct the action αon the infinite tensor product M as above. Define the representations πn : `∞(G) → B(H⊗n) suchthat (πn ⊗ id)(V) = U⊗n.

The strong∗ limitθ(a) := lim

n→∞πn(a)

defines a linear bijection θ between H∞(G, P ) and the relative commutant (Mα)′ ∩M.

Theorem 10.5 probably raises more questions than it answers. First of all, (Mα)′ ∩M is a vonNeumann algebra while H∞(G, P ) has on the first sight no obvious von Neumann algebra structure.Secondly, it is at first sight not clear either how Theorem 10.5 can be useful to really determine(Mα)′ ∩M.

In Section 11 though, we will start from scratch, defining invariant random walks on discretequantum groups and defining their Poisson boundary. Such a Poisson boundary is of the formH∞(G, P ) and carries a natural product making it a von Neumann algebra. Theorem 10.5 thensays that under suitable conditions, the relative commutant (Mα)′∩M of an infinite tensor productaction is isomorphic with the Poisson boundary of an invariant random walk on G.

In Section 12, we shall then study the following questions : can this Poisson boundary be trivial(so that we get a minimal infinite tensor product action on an injective factor) and if not, can wecompute this Poisson boundary ?

Proof of Theorem 10.5. Check as an exercise that

πn+k = (πn ⊗ πk)∆ .

Deduce that En(πn+k(a)) = πn(P k(a)) for all a ∈ `∞(G) and all n, k ∈ N.

So, if a ∈ H∞(G, P ), the sequence πn(a) is a martingale and hence, converges strongly∗ to anelement θ(a) ∈M. For all n and k, we have that πn+k(`∞(G)) commutes with (B(H⊗n))α(n)

insideM. Letting k →∞, θ(a) commutes with (B(H⊗n))α(n)

for all n, meaning that θ(a) ∈ (Mα)′ ∩M.

30

We have shown that θ is a well defined linear map from H∞(G, P ) to (Mα)′∩M. The injectivity ofθ follows from the observation that En(θ(a)) = πn(a) for all n and the assumption that U generatesIrred(G).

To prove the surjectivity of θ, take b ∈ (Mα)′∩M. It follows that En(b) ∈ (B(H⊗n)α(n))′∩B(H⊗n)

for all n. But,B(H⊗n)α(n)

=(πn(`∞(G))

)′so that En(b) ∈ πn(`∞(G)).

Recall that we denoted by px the minimal central projections in `∞(G). We denote by supp(V ) ⊂Irred(G) the set of irreducible representations contained in V . Define

zn =∑

x∈supp(U⊗n)

px

and note that zn equals the support projection of πn as well. So, we can take a unique elementan ∈ zn`∞(G) satisfying En(b) = πn(an). Note also that ‖an‖ ≤ ‖b‖.Since, En(En+k(b)) = En(b), we get that En(πn+k(an+k)) = πn(an), meaning that

znPk(an+k) = an for all n, k .

We claim that there exists a, necessarily unique, element a ∈ H∞(G, P ) such that an = zna for alln. It is then clear that

θ(a) = limnπn(a) = lim

nπn(an) = lim

nEn(b) = b ,

proving the surjectivity of θ.

Because U contains the trivial representation, we get that zn is an increasing sequence of projections.So, it is sufficient to prove that znan+1 = an for all n. But, for all k,

an = znPk+1(an+1+k) and znan+1 = znP

k(an+1+k) .

It follows that ‖an − znan+1‖ ≤ ‖P k+1 − P k‖ ‖b‖ for all k. Again because U contains the trivialrepresentation, we can write P = γP0 + (1− γ)id for some completely positive unital map P0 and0 < γ < 1. From the lemma below, we get that ‖P k+1 − P k‖ → 0 when k → ∞. It follows thatan = znan+1.

We needed (a weak version of) the following lemma, usually referred to as Foguel’s zero-two law.The reason for that name is the following. If A is a commutative von Neumann algebra, theexistence of the S in the lemma is equivalent with ‖P k+1 − P k‖ < 2. Observing that the sequence‖P k+1 − P k‖ is a decreasing sequence, one gets, still for commutative von Neumann algebras A,the following dichotomy : either ‖P k+1 − P k‖ = 2 for all k, either ‖P k+1 − P k‖ converges to 0.

Lemma 10.6 (Ornstein-Sucheston, Foguel, Neshveyev-Tuset). Let A be a unital C∗-algebra andP0 : A→ A a positive linear unital map. Suppose that there exists k ≥ 0 and a positive linear mapS : A→ A such that S(1) is invertible and P k+1 ≥ S and P k ≥ S. Then, ‖Pn+1 − Pn‖ → 0.

Proof. First assume that P = 12(P0 + id) for some linear positive unital map P0. We claim that

‖Pn+1 − Pn‖ → 0. We get

Pn = 2−nn∑

k=0

(n

k

)P k

0 .

31

So,

Pn+1 − Pn = Pn(P − id) =12Pn(P0 − id) = 2−n−1

(−id + Pn+1

0 +n∑

k=1

((n

k − 1

)−

(n

k

))P k

0

).

To get our first claim and because the sequence ‖Pn+1 − Pn‖ decreases, we may assume that n iseven and prove that

2−nn∑

k=1

∣∣∣( n

k − 1

)−

(n

k

)∣∣∣ → 0 . (10.3)

The difference between the absolute value sign at the right hand side is negative when k ≤ n/2 andpositive when k ≥ n/2 + 1. It follows that the expression in (10.3) equals

2−n+1

(n

n/2

)which can be checked to converge to zero by different means.

We now turn to the general case. Our data yield a positive linear map T1 : A→ A satisfying

P k+1 = S1 + P

2+ T1 .

By induction, we find positive linear maps Ta : A→ A satisfying

P a(k+1) = Sa(1 + P

2

)a+ Ta .

Since S(1) is invertible, we have S(1) ≥ ε1 and hence Sa(1) ≥ εa1. So, ‖Ta‖ ≤ 1− εa < 1.

Fix a. By induction, we find positive linear maps Sa,b : A→ A such that

P ab(k+1) = Sa,b

(1 + P

2

)a+ T b

a .

Observe that 0 ≤ T ba(1) ≤ 1 and hence also 0 ≤ Sa,b(1) ≤ 1. So, ‖Sa,b‖ ≤ 1 for all a, b.

Choose ε > 0. By the claim in the beginning of the proof, take a such that∥∥∥(1 + P

2

)a(1− P )

∥∥∥ < ε

2.

Choose b such that ‖T ba‖ ≤ ε/4. We conclude that

‖P ab(k+1)(1− P )‖ < ε .

Since the sequence ‖Pn+1 − Pn‖ = ‖Pn(1− P )‖ is decreasing, the lemma has been proven.

11 Random walks and Poisson boundaries for discrete quantumgroups

11.1 Random walks on discrete groups

Let Γ be a countable group and µ a probability measure on Γ. We define the left-invariant randomwalk on Γ with transition probabilities

p(x, y) = µ(x−1y) .

32

Associated to µ is the Markov operator

P : `∞(Γ) → `∞(Γ) : (P (F ))(x) =∑y∈Γ

F (xy)µ(y)

and one tries to describe all bounded harmonic functions

{F ∈ `∞(Γ) | P (F ) = F} .

As we will see below, there exists a natural probability space (∂Γ, η) with a left action of Γ bybi-measurable, measure class preserving maps on ∂Γ such that the Poisson integral formula

F (x) =∫

∂ΓH(x · ρ) dη(ρ)

establishes a bijection between the set of bounded harmonic functions F and the set of boundedmeasurable functions H on (∂Γ, η). One calls (∂Γ, η) with its left Γ-action, the Poisson boundaryof the random walk.

Although Γ y (∂Γ, η) is abstractly constructed below, it is next an interesting problem to takeconcrete groups Γ, concrete transition probabilities µ and to try to identify Γ y (∂Γ, η) with someknown action of Γ.

11.2 Random walks on discrete quantum groups

Fix a compact quantum group G and denote by Irred(G) the set of equivalence classes of irreducibleunitary representations. Choose for every x ∈ Irred(G) a representative Ux ∈ B(Hx)⊗ C(G) withUx being a unitary representation of G on Hx. Recall that

`∞(G) =∏

x∈Irred(G)

B(Hx) and V ∈ M(c0(G)⊗ C(G)) : V =⊕

x∈Irred(G)

Ux .

The comultiplication ∆ : `∞(G) → `∞(G)⊗`∞(G) is defined such that (∆⊗ id)(V) = V13V23.

We denote by ε the one-dimensional representation of `∞(G) given by the trivial representation ofG.

Analogous to Definition 9.1, we define actions of discrete quantum groups :

Definition 11.1. A (left) action of the discrete quantum group G on the von Neumann algebra Mis a faithful normal unital ∗-homomorphism α : M → `∞(G)⊗M satisfying (∆⊗ id)α = (id⊗ α)α.

Finally, we use V to define the so-called adjoint action of G on `∞(G).

αG : `∞(G) → `∞(G)⊗L∞(G) : αG(a) = V(a⊗ 1)V∗ . (11.1)

Note that the fixed point algebra of αG is exactly the center of `∞(G).

Notation 11.2. Whenever φ is a normal state on `∞(G), we define the Markov operator

Pφ : `∞(G) → `∞(G) : Pφ(a) = (id⊗ φ)∆(a) .

33

Because of Proposition 11.3 below, we will not be interested in taking arbitrary states φ. Recallfrom (4.2) that we can define faithful states ψx on B(Hx) by the formula

ψx(T )1 = (id⊗ h)(Ux(T ⊗ 1)(Ux)∗) .

Denote by Qx the positive invertible matrix in B(Hx) satisfying

Tr(Qx) = Tr(Q−1x ) and (Q1/2

x ⊗ 1)U(Q−1/2x ⊗ 1) unitary.

Recall from (4.2) that

ψx(T ) =Tr(QxT )Tr(Qx)

.

Having at hand the specific state ψx on B(Hx), we now define for any probability measure µ onIrred(G),

ψµ =∑

x∈Irred(G)

µ(x)ψx and Pµ = (id⊗ ψµ)∆ .

We also write Px = (id⊗ ψx)∆ and put

H∞(G, µ) := {a ∈ `∞(G) | Pµ(a) = a} .

Proposition 11.3. Let φ be a normal state on `∞(G). Then the following are equivalent.

1. The Markov operator Pφ preserves the center of `∞(G).

2. There exists a probability measure µ on Irred(G) such that φ = ψµ.

3. The state φ is invariant under the adjoint action αG defined in (11.1) : (φ⊗ id)αG(a) = φ(a)1for all a ∈ `∞(G).

Before proving Proposition 11.3, one needs to make the following exercise.

Exercise 24. Denote for all x, y, z ∈ Irred(G) by px⊗yz the orthogonal projection of Hx ⊗Hy onto

the sum of all subspaces on which UxT© Uy is unitarily equivalent with U z. Prove that

(px ⊗ py)∆(pz) = px⊗yz .

In order to do so, it is sufficient to check that left-hand side and right-hand side are equal aftermultiplication on the right by an arbitrary T ∈ Mor(Ux

T© Uy, U r).

Proof. First note that 2 and 3 follow equivalent because of (4.2) saying that

ψx(T )1 = (id⊗ h)(Ux(T ⊗ 1)(Ux)∗) for all T ∈ B(Hx) .

Suppose next that 1 holds. Denoting by ε the trivial representation of G, Exercise 24 implies that

(px ⊗ 1)∆(pε) = px⊗xε .

On the other hand, Theorem 4.7 tells us that

px⊗xε = ss∗ where s = Tr(Qx)−1/2

∑i

Q−1/2x ei ⊗ ei .

34

Since Pφ preserves the center of `∞(G), it follows that (id ⊗ φ)(ss∗) is a multiple of px for allx ∈ Irred(G). Filling in the concrete formula for s, we get that the restriction of φ to B(Hx) is amultiple of ψx for all x. This yields 2.

To prove that 2 implies 1, it suffices to check that px(id ⊗ ψy)∆(pz) is a multiple of px for allx, y, z ∈ Irred(G). Choosing a unit invariant vector t ∈ Hy ⊗Hy, we get that

px(id⊗ ψy)∆(pz) = (1⊗ t∗)(px⊗yz ⊗ 1)(1⊗ t) .

But the right-hand side belongs to End(Ux) = Cpx.

So, if Pµ is the Markov operator defined by the probability measure µ on Irred(G), the restriction ofPµ to the center of `∞(G) defines an ordinary random walk on Irred(G) with transition probabilitiesgiven by

pxp(x, y) = pxPµ(py)

where we recall that the px are the minimal central projections in `∞(G). Using the same formulasas in Exercise 24 and the proof of Proposition 11.3, one can check that for all x, y ∈ Irred(G),

p(x, y) =∑

z∈Irred(G)

µ(z) mult(y, x T© z) dimq(y)dimq(x) dimq(z)

where dimq(x) = Tr(Qx) = Tr(Q−1x ) denotes the quantum dimension of Ux. We also used the

short-hand notation mult(y, x T© z) = mult(Uy, UxT© U z).

11.3 Abstract construction of the Poisson boundary

From now on, we will restrict our attention to generating probability measures µ, meaning thatthe support of µ generates Irred(G), i.e. every irreducible representation of G appears in a tensorproduct of irreducible representations in the support of µ.

In fact, one can define a convolution product on the probability measures on Irred(G) such that

ψµ∗η = (ψµ ⊗ ψη)∆ .

Then, the support of µ∗n consists of those irreducible representations contained in an n-fold tensorproduct of elements of suppµ.

Proposition 11.4. Let G be a compact quantum group and let µ be a generating probability measureon Irred(G). For all a, b ∈ H∞(G, µ), the sequence Pn

µ (ab) is strong∗-convergent with limit containedin H∞(G, µ).

• Equipped with the product a · b := limn Pnµ (ab), H∞(G, µ) is a von Neumann algebra.

• The restriction of ε to H∞(G, µ) defines a faithful normal state on H∞(G, µ).

• The restriction of ∆ defines a left action αbG : H∞(G, µ) → `∞(G)⊗H∞(G, µ) of G on

H∞(G, µ).

• The restriction of the adjoint action αG defines the right action αG : H∞(G, µ) → H∞(G, µ)⊗L∞(G)of G on H∞(G, µ).

35

• The Poisson integral formula a = (id ⊗ ε)∆(a) defines the (tautological, identity) bijectivelinear map between H∞(G, µ) and H∞(G, µ) !

Certainly the last item of Proposition 11.4 is rather disappointing. This is not our dreamed wayof representing bounded harmonic functions. In concrete examples it is an often difficult task toidentify the von Neumann algebra H∞(G, µ) and the action of G, with some known action on someknown von Neumann algebra. Only then, we can really describe the bounded harmonic functions.

Proof of Proposition 11.4. We first prove the most non-trivial statement : whenever a, b ∈ H∞(G, µ),the sequence Pn

µ (ab) converges ∗-strongly and the formula a · b = limn Pnµ (ab) defines a product on

H∞(G, µ) in such a way that H∞(G, µ) becomes a von Neumann algebra.

Define H =⊕

x∈supp µHx and U =⊕

x∈supp µ Ux. Take the unique normal faithful state ω on B(H)

that is invariant under the inner action of G on B(H) defined by U and that satisfies ω ◦ π = ψµ,where π : `∞(G) → B(H) is such that U = (π ⊗ id)(V).

Define the infinite tensor product von Neumann algebra

M =⊗∞

n=1(B(H), ω) .

Define πn : `∞(G) → B(H⊗n) : (πn ⊗ id)(V) = U⊗n. Whenever a ∈ H∞(G, µ), the sequence πn(a)is a martingale and hence, converges ∗-strongly to an element π∞(a) ∈ M. We have obtained thelinear map

π∞ : H∞(G, µ) →M .

Whenever a, b ∈ H∞(G, µ), we have

En(π∞(a)π∞(b)) = limkEn(πn+k(a)πn+k(b)) = lim

kEn(πn+k(ab)) = lim

kπn(P k(ab)) .

Denoting by zn the support projection of πn, we conclude that for fixed n, the sequence (znP k(ab))k

is ∗-strongly convergent. Since µ is a generating probability measure and the sequence (P k(ab))k

is bounded, it follows that (P k(ab))k is ∗-strongly convergent. Denoting its limit by a · b, we get byconstruction π∞(a · b) = π∞(a)π∞(b).

Again because µ is generating, π∞ is isometric. We conclude that the new product on H∞(G, µ)turns H∞(G, µ) into a ∗-algebra in such a way that π∞ is a ∗-homomorphism. But, π∞ is weaklycontinuous on the (weakly compact) unit ball of H∞(G, µ) implying that the image of π∞ is a vonNeumann subalgebra of M. This proves the first statement.

If a is a positive element in H∞(G, µ) and ε(a) = 0, it follows that also ε(P kµ (a)) = 0 for all k.

This means that ψµ∗k(a) = 0 for all k. Since µ is generating, it follows that a = 0, proving thefaithfulness of ε.

The remaining statements follow from the following equivariance formulas :

∆ ◦ Px = (id⊗ Px) ◦ ∆ and (Px ⊗ id) ◦ α = α ◦ Px for all x ∈ Irred(G) .

The first of these covariance formulas follows immediately from the coassociativity of ∆. The secondone follows from the following computation employing the invariance of ψx under the adjoint action :

(Px ⊗ id)α(a) = (id⊗ ψx ⊗ id)(∆⊗ id)(V(a⊗ 1)V∗)

= (id⊗ ψx ⊗ id)(V13 V23 (∆(a)⊗ 1) V∗23 V∗

13)

= V(id⊗ ψx ⊗ id)((1⊗ Ux)(∆(a)⊗ 1)(1⊗ (Ux)∗)

)V∗

= V((id⊗ ψx)∆(a)⊗ 1

)V = α(Px(a)) .

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Remark 11.5. Let G be a compact quantum group with generating unitary representation Ucontaining the trivial representation. As in the previous section, we take a faithful normal stateω on B(H), invariant under the inner action of G on B(H) defined by U . We define the infinitetensor product action α of G on M =

⊗n(B(H), ω).

Take π : `∞(G) → B(H) such that (π⊗ id)(V) = U . It follows that ω ◦π = ψµ for some probabilitymeasure µ on Irred(G). Hence, Theorem 10.5 implies that the relative commutant (Mα)′ ∩M isisomorphic with the Poisson boundary H∞(G, µ).

12 Computations of Poisson boundaries

The first result that we prove, is a rather amazing one : if the compact quantum group G is not ofKac type, its Poisson boundary is never trivial.

Theorem 12.1 (Izumi [7]). Let G be a compact quantum group that is not of Kac type. Let µ beany generating probability measure on Irred(G). Then, H∞(G, µ) is non-trivial. In particular, theinfinite tensor product actions of G constructed in Exercise 23 are never minimal.

Proof. Consider Izumi’s operator

Θ : L∞(G) → `∞(G) : Θ(a) = (id⊗ h)(V∗(1⊗ a)V) .

Claim 1. The image of Θ is invariant under all Markov operators, i.e.

Px(Θ(a)) = Θ(a) for all x ∈ Irred(G), a ∈ L∞(G) .

Claim 2. The image of Θ equals C1 if and only if G is of Kac type.

To prove claim 1, observe that (5.1) and the formula ψx(T ) = Tr(QxT )/Tr(Qx) imply that

(ψx ⊗ h)(aUx) = (ψx ⊗ h)((Q−2x ⊗ 1)Uxa) for all a ∈ B(Hx)⊗ C(G) .

Writing ϕx(T ) = Tr(Q−1x T )/Tr(Qx), we observe that ϕx is still a state on B(Hx) and that

(ψx ⊗ h)(aUx) = (ϕx ⊗ h)(Uxa) for all a ∈ B(Hx)⊗ C(G) .

We are now ready to prove claim 1 :

Px(Θ(a)) = (id⊗ ψx ⊗ h)(∆⊗ id)(V∗(1⊗ a)V)= (id⊗ ψx ⊗ h)

((Ux

23)∗V∗

13(1⊗ 1⊗ a)V13Ux23

)= (id⊗ ϕx ⊗ h)

(V∗

13(1⊗ 1⊗ a)V13

)= Θ(a) .

It remains to prove claim 2. If G is of Kac type, the equality Θ(a) = h(a)1 follows by using that his a trace and U a unitary for all irreducible representations U .

If we assume conversely that the image of Θ equals C1, it follows that Θ(a) = ψx(Θ(a))1. Thesame formula as above yields ψx(Θ(a)) = h(a) and so

(id⊗ h)((Ux)∗(1⊗ a)Ux) = h(a)1 for all a ∈ L∞(G), x ∈ Irred(G) . (12.1)

37

Using first (4.2) and applying next (12.1) to x, we get for all x ∈ Irred(G) and all T ∈ B(Hx),

ψx(T )1⊗ 1 = 1⊗ (id⊗ h)(Ux(T ⊗ 1)(Ux)∗) = (id⊗ id⊗ h)(Ux13U

x23(1⊗ T ⊗ 1)(Ux

23)∗(Ux

13)∗) .

Let now s be an invariant unit vector inHx⊗Hx. Putting s∗ and s around the previous computation,it follows that

ψx(T ) = s∗(1⊗ T )s = ϕx(T ) for all x ∈ Irred(G), T ∈ B(Hx) .

So, we get that Qx = 1 for all x and hence, G is of Kac type.

If on the other hand G is of Kac type and is co-amenable, then the Poisson boundary can be trivial.The precise statement goes as follows.

Theorem 12.2 (Kaimanovich-Vershik, V [10]). Let G be a co-amenable compact quantum groupof Kac type. Then there exists a generating probability measure µ on Irred(G) such that H∞(G, µ)is trivial. In particular, G admits a minimal action on the hyperfinite II1 factor.

Proof. See Lemma 7.1 in [10].

12.1 Work of Tomatsu : co-amenable compact quantum groups with commu-tative fusion rules

In his recent article [9], Tomatsu has computed the Poisson boundary for a quite large class of com-pact quantum groups. He deals with all co-amenable compact quantum groups having commutativefusion rules. As we will see, this is a purely quantum work : if the fusion rules are commutative,the Poisson boundary of the ordinary random walk on Irred(G) is trivial.

In order to state Tomatsu’s theorem and to give a sketch of its proof, we need quite a bit of materialthat we gathered in three paragraphs below :

• unital completely positive maps and their multiplicative domain,

• the notion of a closed quantum subgroup K of G and the corresponding homogeneous spaceL∞(K\G),

• the maximal closed quantum subgroup of Kac type.

When G is a compact quantum group with closed quantum subgroup K, the restriction of thecomultiplication ∆ defines a right action of G on K\G that we denote by

βG : L∞(K\G) → L∞(K\G)⊗L∞(G) . (12.2)

We also denote by βbG the adjoint action of G on L∞(G) defined by

βbG : L∞(G) → `∞(G)⊗L∞(G) : β

bG(a) = V∗(1⊗ a)V . (12.3)

38

Theorem 12.3 (Tomatsu [9]). Let G be a co-amenable compact quantum group having commutativefusion rules and let µ be a generating probability measure on Irred(G). Let K be the maximal closedquantum subgroup of Kac type. Then, the multiplicative domain of the Izumi operator

Θ : L∞(G) → H∞(G, µ) : Θ(a) = (id⊗ h)(V∗(1⊗ a)V) (12.4)

equals L∞(K\G) and the restriction of Θ defines a ∗-isomorphism

Θ : L∞(K\G) → H∞(G, µ) .

Moreover, Θ is G-equivariant and G-equivariant, meaning that

αG ◦Θ = (Θ⊗ id) ◦ βG and αbG ◦Θ = (id⊗Θ) ◦ β

bG .

Proof. Fix any generating probability measure µ on Irred(G). The commutativity of the fusionrules implies the following.

• (Kaimanovich, Hayashi, see Corollary 3.2 in [8].)The intersection H∞(G, µ) ∩ Z(`∞(G)) is trivial. In words : the Poisson boundary of theassociated random walk on the set Irred(G) is trivial.

• (Izumi, Neshveyev, Tuset, see Proposition 1.1 in [8].)The Poisson boundary H∞(G, µ) is independent of the choice of generating measure µ, i.e.

H∞(G, µ) = {a ∈ `∞(G) | Px(a) = a for all x ∈ Irred(G)} .

The co-amenability of G implies the amenability of G which means that there exists a (in generalnon-normal) state m ∈ `∞(G) satisfying

(id⊗m)∆(a) = m(a)1 for all a ∈ `∞(G) .

Since m is not normal in general, one has to be careful with the definition of

id⊗m : M⊗`∞(G) →M

wheneverM is a von Neumann algebra. In fact, id⊗m is defined in such a way that ω((id⊗m)(a)) =m((ω ⊗ id)(a)) for all a ∈M⊗`∞(G) and all ω ∈M∗, making use of the fact that M = (M∗)∗.

Using m, we define a completely positive unital map

Ψ : H∞(G, µ) → L∞(G) : Ψ(a) = (m⊗ id)αG(a) ,

where αG : H∞(G, µ) → H∞(G, µ)⊗L∞(G) : αG(a) = V(a ⊗ 1)V∗ denotes as before the adjointaction of G on H∞(G, µ). Note that Ψ is possibly not a normal map.

Computation 1. The completely positive map Ψ is G-equivariant : βbG ◦Ψ = (id⊗Ψ) ◦ α

bG.

βbG(Ψ(a)) = V∗(1⊗Ψ(a))V

= V∗(id⊗m⊗ id)(∆⊗ id)(V(a⊗ 1)V∗)= (id⊗Ψ)α

bG(a) .

39

Computation 2. The conditional expectation

E : `∞(G) → Z(`∞(G)) : E(a) = (id⊗ h)αG(a)

satisfies Px ◦ E = E ◦ Px for all x ∈ Irred(G). Indeed, using the invariance of ψx under the inneraction defined by Ux, we get

Px(E(a)) = (id⊗ ψx ⊗ h)(∆⊗ id)(V(a⊗ 1)V∗)

= (id⊗ ψx ⊗ h)(V13V23(∆(a)⊗ 1)V∗23V∗

13)= (id⊗ h)(V(Px(a)⊗ 1)V∗) = E(Px(a)) .

As we observed above, the Poisson boundary H∞(G, µ) intersects trivially the center of `∞(G).Combined with computation 2, it follows that E(a) = ε(a)1 for all a ∈ H∞(G, µ).

Computation 3. We have h◦Ψ = ε and so, Ψ : H∞(G, µ) → L∞(G) is a faithful, normal, unital,completely positive map. Indeed, h(Ψ(a)) = m(E(a)) = ε(a).

Computation 4. We have Θ(Ψ(a)) = a for all a ∈ H∞(G, µ). Using computations 1 and 3, wehave

Θ(Ψ(a)) = (id⊗ h)βbG(Ψ(a)) = (id⊗ h ◦Ψ)α

bG(a) = (id⊗ ε)αbG(a) = a .

Claim. Denoting by M the multiplicative domain of Θ, the restriction of Θ is a ∗-isomorphismM → H∞(G, µ) with inverse Ψ. By definition, Θ is a faithful normal ∗-homomorphism from Minto H∞(G, µ). For all a ∈ H∞(G, µ), we have

a∗a = Θ(Ψ(a))∗Θ(Ψ(a)) ≤ Θ(Ψ(a)∗Ψ(a)) ≤ Θ(Ψ(a∗a)) = a∗a .

It follows that Ψ(a) ∈M with Θ(Ψ(a)) = a. But then Θ : M → H∞(G, µ) is a ∗-isomorphism withinverse Ψ, proving the claim.

Computation 5. The completely positive map Θ : L∞(G) → H∞(G, µ) is G-equivariant, i.e.(Θ⊗ id) ◦∆ = αG ◦Θ.

αG(Θ(a)) = V(Θ(a)⊗ 1)V∗ = V(id⊗ h⊗ id)(id⊗∆)(V∗(1⊗ a)V)V∗

= (id⊗ h⊗ id)(V∗12(1⊗∆(a))V) = (Θ⊗ id)∆(a) .

Computation 6. The completely positive map Θ : L∞(G) → H∞(G, µ) is G-equivariant, i.e.(id⊗Θ) ◦ β

bG = αbG ◦Θ.

αbG(Θ(a)) = ∆(Θ(a)) = (id⊗ id⊗ h)(V∗

23V∗13(1⊗ 1⊗ a)V13V23)

= (id⊗Θ)(V∗(1⊗ a)V) = (id⊗Θ)βbG(a) .

Let a ∈M . Computation 5 implies that ∆(a) belongs to the multiplicative domain of Θ⊗ id. Butthis implies that

(Θ⊗ id)(∆(a)(b⊗ 1)) = (Θ⊗ id)∆(a) (Θ(b)⊗ 1)

for all b ∈ L∞(G). It follows that ∆(M) ⊂M⊗L∞(G). An analogous reasoning using computation6 implies that β

bG(M) ⊂ `∞(G)⊗M .

Because of computations 3 and 4, the map Ψ◦Θ is the h-preserving normal conditional expectationof L∞(G) onto M . The G-equivariance of both Θ and Ψ (computations 1 and 6) imply that Ψ ◦Θ

40

is G-equivariant. Theorem 12.8 gives us a closed quantum subgroup K of Kac type such thatM = L∞(K\G).

The theorem is proved once we argue that K is maximal. In order to obtain this last step, it issufficient to take an arbitrary closed quantum subgroup K1 of Kac type with conditional expectationEK1 : L∞(G) → L∞(K1\G) and to make the following

Computation 7. Θ(a) = Θ(EK1(a)) for all a ∈ L∞(G).Indeed, this then implies that L∞(K\G) ⊂ L∞(K1\G). Denote by h1 the Haar state of K1. As inthe proof of Theorem 12.1, we know that (id ⊗ h1)(U∗(1 ⊗ a)U) = h1(a)1 for all a ∈ C(K1). Wenow compute for all a ∈ Calg(G) :

Θ(a) = (id⊗ h1π ⊗ h)(id⊗∆)(V∗(1⊗ a)V)= (id⊗ h1π ⊗ h)(V∗

13V∗12(1⊗∆(a))V12V13)

= (id⊗ h)(V∗(id⊗ h1π ⊗ id)(V∗12(1⊗∆(a))V12)V)

= (id⊗ h)(V∗(1⊗ (h1π ⊗ id)∆(a))V)= Θ(EK1(a)) .

This ends the proof of the theorem.

The easiest example of a compact quantum group satisfying the conditions of Theorem 12.3 isG = SUq(2) for q ∈ [−1, 1] \ {0}. When q = ±1, G is of Kac type and Theorem 12.3 says that thePoisson boundary H∞(G, µ) is trivial for every generating measure µ.

If q 6= ±1, we already knew from Theorem 12.1 that the Poisson boundary H∞(G, µ) is non-trivial.It is not hard to check that the maximal closed quantum subgroup of Kac type is the maximaltorus T defined by the ∗-homomorphism

π : Calg(G) → C(T) : (id⊗ π)(U) =(z 00 z

)where U denotes the fundamental representation of G = SUq(2) and z denotes the obvious functiongenerating C(T).

Theorem 12.3 now tells us that for any generating measure µ, the Poisson boundary H∞(G, µ) isgiven by the homogeneous space

H∞(G, µ) ∼= L∞(T\SUq(2)) .

The quantum homogeneous space T\SUq(2) with its natural action of SUq(2) is usually calledPodles’ sphere.

Remark 12.4. The Poisson boundary for the dual of SUq(2) has first been computed by Izumi in[7], using a much more elaborate argument. Based on [7], Izumi, Neshveyev and Tuset computedthe Poisson boundary for the dual of SUq(n) in [8], still identifying it with a quantum homogeneousspace of the form Tn−1\SUq(n). The simplest approach is the most recent one due to Tomatsupresented above. It covers all q-deformations of the classical compact Lie groups.

Unital completely positive maps and their multiplicative domain

Theorem 12.5 (Choi). Let A,B be unital C∗-algebras and P : A→ B a unital completely positive(ucp) map.

41

1. For all a ∈ A, we have P (a)∗P (a) ≤ P (a∗a).

2. Let a ∈ A. Then, P (a)∗P (a) = P (a∗a) if and only if P (ba) = P (b)P (a) for all b ∈ A.

3. Let a ∈ A. Then, P (a)P (a)∗ = P (aa∗) if and only if P (ab) = P (a)P (b) for all x ∈ A.

4. The set D := {a ∈ A | P (a)∗P (a) = P (a∗a) , P (a)P (a)∗ = P (aa∗)} is a unital C∗-subalgebraof A, called multiplicative domain of P . The restriction of P to D is a unital ∗-homomorphismfrom D to B.

Proof. Representing B faithfully on a Hilbert space, we can assume from the beginning that B =B(H). To prove 1, we first construct the Stinespring dilation of P , yielding a Hilbert space K, aunital ∗-homomorphism π : A → B(K) and an isometry V : H → K such that P (a) = V ∗π(a)Vfor all a ∈ A.

Exercise 25.

• Prove that the formula〈a⊗ ξ, b⊗ η〉 := 〈ξ, P (a∗b)η〉

defines a positive scalar product on the vector space A ⊗alg H. Define K as its separa-tion/completion and define the isometry V : H → K : V (ξ) = 1⊗ ξ.

• Prove that the formula π(a)(b⊗ ξ) = ab⊗ ξ extends to a well-defined unital ∗-homomorphismπ : A→ B(K).

• Prove that P (a) = V ∗π(a)V for all a ∈ A.

With the exercise at hand, we immediately get P (a)∗P (a) = V ∗π(a)∗V V ∗π(a)V ≤ V ∗π(a)∗π(a)V =P (a∗a).

We see at the same time that if P (a)∗P (a) = P (a∗a), then V ∗π(a)∗(1− V V ∗)π(a)V = 0. Writingy = (1− V V ∗)1/2π(a)V , this means that y∗y = 0 and hence, y = 0. But then, for all b ∈ A,

V ∗π(b)(1− V V ∗)π(a)V = V ∗π(b)(1− V V ∗)1/2y = 0

yielding 2. Of course, 3 follows by symmetry. Finally, using 2 and 3, statement 4 is obvious.

Closed quantum subgroups and their homogeneous spaces

Definition 12.6. A closed quantum subgroup of a compact quantum group G is a compact quantumgroup K together with a surjective ∗-homomorphism

π : Calg(G) → Calg(K)

satisfying ∆ ◦ π = (π ⊗ π) ◦∆.

It is important to observe that some authors, including Tomatsu, use a more restrictive definitionof a closed quantum subgroup. For the following reasons, we believe that the above definition isthe most natural one :

• when G is a compact group, we retrieve the ordinary notion of a closed subgroup,

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• when G = Γ, the closed quantum subgroups of G are given as Γ1 for Γ1 a quotient of Γ,

• the trivial compact quantum group is always a closed quantum subgroup through the co-unitε : Calg(G) → C.

Notation 12.7. Let G be a compact quantum group with closed quantum subgroup K defined byπ : Calg(G) → Calg(K). We denote

Calg(K\G) := {a ∈ Calg(G) | (π ⊗ id)∆(a) = 1⊗ a} .

We define L∞(K\G) as the weak closure of Calg(K\G) in B(L2(G)).

Formulas (12.2) and (12.3) define actions βG of G and βbG of G on L∞(K\G).

The map (hKπ⊗ id)∆ extends to the (necessarily unique) Haar state preserving conditional expec-tation

EK : L∞(G) → L∞(K\G) .

The following is one of the main results of [9] and a central ingredient of the proof of Theorem 12.3above. The result tells us which von Neumann subalgebras M ⊂ L∞(G) are of the form L∞(K\G).

Theorem 12.8 (Tomatsu [9]). Let G be a compact quantum group and M ⊂ L∞(G) a von Neumannsubalgebra. Recall the adjoint action β

bG of G on L∞(G) defined by (12.3). Then, the following areequivalent.

• There exists a closed quantum subgroup K of G such that M = L∞(K\G).

• The following conditions hold.

– ∆(M) ⊂M⊗L∞(G).

– βbG(M) ⊂ `∞(G)⊗M .

– There exists a Haar state preserving conditional expectation E : L∞(G) →M .

Moreover, K is of Kac type if and only if the conditional expectation EK is G-equivariant, i.e.(id⊗ EK)β

bG(a) = βbG(EK(a)) for all a ∈ L∞(G).

We even do not sketch the proof of this theorem and refer to [9].

The maximal closed quantum subgroup of Kac type

The object in the title is characterized by the following result.

Proposition 12.9 (V). Let G be a compact quantum group. Up to isomorphism, there is a uniqueclosed quantum subgroup K of G defined by π : Calg(G) → Calg(K) having the following properties.

• K is of Kac type.

• Whenever K1 is a closed quantum subgroup of G, of Kac type and defined by the surjective∗-homomorphism π1 : Calg(G) → Calg(K1), there exists a ∗-homomorphism θ : Calg(K) →Calg(K1) satisfying π1 = θ ◦ π.

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Sketch of proof. The only issue is to prove the existence of K. Define the closed 2-sided ideal I ofCu(G) as

I := {a ∈ Cu(G) | τ(a∗a) = 0 for all tracial states τ on Cu(G) } .

Define the unital C∗-algebra C(K) := Cu(G)/I and denote by π the quotient map.

Whenever τ1, τ2 are tracial states on Cu(G), (τ1 ⊗ τ2)∆ is again a tracial state. Therefore, there isa well-defined unital ∗-homomorphism ∆ : C(K) → C(K) ⊗ C(K) satisfying ∆ ◦ π = (π ⊗ π) ◦∆.It is easy to check that (C(K),∆) is again a compact quantum group. Moreover, by constructionC(K) has a separating family of tracial states. Repeating the existence proof of the Haar state ona compact quantum group and systematically replacing ‘state’ by ‘tracial state’, it follows that theHaar state of K is a trace. Therefore K is of Kac type.

Since the Haar state of any other closed quantum subgroup of Kac type is a tracial state, themaximality condition in the proposition is easily checked to hold for K.

12.2 Poisson boundaries and monoidal equivalence

In [5] the notion of monoidal equivalence of compact quantum groups was studied in detail andrelated to the study of ergodic actions of compact quantum groups. Note that an action α : M →M⊗L∞(G) of a compact quantum group on a von Neumann algebra M is called ergodic if thefixed point algebra Mα equals C1.

It would lead us too far to introduce the machinery of monoidal equivalence, but for completeness,I want to state in a very unprecise way the following recent result by De Rijdt and Vander Vennet.

Theorem 12.10 (De Rijdt - Vander Vennet [6]). Let G1 and G2 be monoidally equivalent compactquantum groups. This means in particular that there is a bijection between Irred(G1) and Irred(G2)preserving the fusion rules (but that alone is not sufficient). Then, given a generating measure µ onIrred(G1) (and hence on Irred(G2) through this bijection), one can describe the Poisson boundaryH∞(G2, µ) in terms of H∞(G1, µ).

So, in order to apply Theorem 12.10, we need to know which compact quantum groups aremonoidally equivalent. Some examples are treated in [5], proving that if Fi ∈ GLni(C) satisfyFiF i = ci1 with ci = ±1, then the quantum groups Ao(F1) and Ao(F2) are monoidally equivalentif and only if c1 = c2 and Tr(F ∗1F1) = Tr(F ∗2F2).

In particular, for every F ∈ GLn(C), there is a unique q ∈ [−1, 1] \ {0} such that Ao(F ) ismonoidally equivalent with SUq(2). As a result, one can combine Theorems 12.3 and 12.10 todescribe the Poisson boundary for the dual of Ao(F ).

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