Poisson and Martin boundaries of discrete quantum groups: a noncommutative and categorical perspective

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Poisson and Martin boundaries of discrete quantum groups: a noncommutative and

categorical perspective

Sara Malacarne

Thesis submitted for the degree of Ph.D.

Department of Mathematics University of Oslo

September 2018

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Acknowledgements

These PhD years here in Oslo have been one of the best times of my life. The challenges and hard work made my achievements even more rewarding, but I couldn’t have accomplished what I have without the support of some people whom I wish to thank here.

I would like to start by thanking my supervisor, Sergey Neshveyev. Not only for the stimulating discussions or for the inspiration during frustrating times when the finishing line seemed so far away. Most importantly for the passion for mathematics that was passed on to me and for motivating me to do my best.

I want to express my gratitude to Florin Radulescu, my co-supervisor, who pointed out the great opportunity of taking a PhD in Oslo to me.

Thank you to the Operator Algebras group, to the ones that are still part of it and to the ones that have left, who contributed to making the atmosphere at the department very pleasant and enjoyable, especially during the 3 o’clock coffee breaks. I am in particular very grateful to Adam, Marco, Bas and Magdalena whom I met during my first year in Oslo.

We started by discussing about mathematics during seminars and ended up building a lovely friendship.

A big thank you to the wonderful friends I met in Oslo and to the Gigogin, my "norwe- gian family". I would like to mention Valentina and Valeria for their closeness during tougher times.

I am deeply thankful to my parents Mauro and Carolyn, to my brothers Luca and Andrea and to Alessandra for always believing in me.

I would like to end by thanking my husband Francesco for his unconditional love and support. There hasn’t been a moment he wasn’t by my side encouraging me.

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Introduction

This thesis principally focuses on probabilistic boundaries of duals of compact quantum groups. We here give a brief historical introduction in order to understand how these mathematical objects appeared and what connections they have with the field of operator algebras.

The notion of a noncommutative space originated during the developments in quantum physics. Informally speaking, to get a more accurate description of nature at the atomic-scale, it was realised that the real-valued functions describing a physical system had to be replaced by noncommutative operators on a Hilbert space, see for example [75], [77]. This is based on the fact that physical quantities of a quantum system can only be measured with finite precision and the process of measuring, simultaneously perturbs the system. The aforemen- tioned discussion is summarised by the Heisenberg uncertainty relations and it implies that it matters in which order the measurement of physical quantities, such as position q and momentum p, is performed. This brings us to Heisenberg’s commutation relations

[p,q] = h 2πi,

where h is Planck’s constant. Thus, quantum mechanics can be viewed as a deformation of classical mechanics, the deformation parameter being Planck’s constant h, and classical mechanics corresponding toh→0.

A physical system is described by states and observables. Observables are the physical quantities such as position, momentum, energy, and so on, that one is interested in studying and which can be measured. The state of a system is a specification of the condition of the system from which one can deduce information about the observables of interest; the set of all possible states is called the state space. Using a mathematical formulation, in classical mechanics states are points of a manifold while observables can be assumed to be real-valued smooth functions defined on it. When passing from classical to quantum mechanics it turns out that the observables generate a non abelian C^∗-algebra and the states of the system are the positive linear functionals [75]. The noncommutative algebra of observables can still be thought of as an algebra of continuous functions, but on a noncommutative space.

This led to the discovery of a new notion of geometry, generally known as noncommutative geometry, developed by Connes in [16], who realised that the classical notion of a space imposed limitations to the theory. In fact by the Gelfand theorem, according to which commutativeC^∗-algebras can be identified with algebras of continuous functions on a locally compact Hausdorff space, a space could equivalently well be studied by its cooordinate func-

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tions rather than its points. The same duality philosophy was also adopted by Voiculescu and subsequently by Speicher, Nica and many others, for the development of free probability, a noncommutative probability theory [94], [67].

Depending on the characteristics of the algebra of functions, one can study different properties of a space. In the same way, if one considersC^∗-algebras, the attention is given to the topological aspects of the space, while if working with von Neumann algebras the measure- theoretical aspects are highlighted.

Replacing a space by a group, one can ask whether its properties are still recovered from the algebra of functions on it. This relates to Hopf’s work on algebraic topology [31] . Given a compact Lie groupGand its cohomology ringH^∗(G), Hopf investigated the properties of the comultiplication map ∆:H^∗(G) → H^∗(G)⊗H^∗(G) induced by the multiplication on the group and the corresponding structure endowed on H^∗(G). Hopf algebras were then defined but there were no purely noncommutative examples, all examples being in some way connected to classical groups. It was in the work of Kulish, Reshetikhin, Sklyanin, Takhtajan and Faddeev who were interested in quantum inverse scattering problems that the first quantum groups appeared, even though not recognised as such. Finally, Drinfeld [23]

and Jimbo [37] definedq-deformed Hopf algebras for every complex semisimple Lie algebra.

The meeting point between Hopf algebra theory and operator algebras emerged from the work of Kac [41] and was fully developed by Kac and Vainerman [86] and Enock and Schwartz [25] who aimed to generalise the Pontrjagin duality for abelian locally compact groups. The idea was to work with von Neumann algebras instead of Hopf algebras and study the properties of the comultiplication map in this setting. AC^∗-algebraic counterpart was established by Vallin and Enock [87], [26]. Kac algebras were introduced, but again, interesting examples were left out since the assumptions were too restrictive. In [101], Woronow- icz systematically defined compact matrix pseudogroups, that is, universal unitalC^∗-algebras generated by the entries of a matrix(u_{i j})i,j with comultiplication map∆given by

∆(u_{i j}) =X

k

u_{i k}⊗u_{k j},

and satisfying certain properties. Assuming the C^∗-algebra to be the algebra of continuous functions on a group, these properties would allow to completely recover the group structure. In addition, in [101] Woronowicz also discussedSU_q(2)in theC^∗-algebraic setting. The term "quantum group" was suggested by Drinfeld and denoted the deformation of the universal enveloping algebra of the complex semisimple Lie algebra. Nowadays, inspired by the noncommutative geometry language, one usually calls a quantum group the virtual geometric object corresponding to a noncommutativeC^∗-algebra equipped with a comultiplication map. Later in [103], a comultiplication map was defined for general, not necessarily finitely generated, unitalC^∗-algebras, thus, obtaining a complete axiomatisation of compact quantum groups in the operator algebraic setting.

Many concepts in the classical theory of compact groups are extended to compact quantum groups and the existence of a Haar measure in this setting is proved. TheC^∗-algebras have

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to be assumed to be unital. Dropping this assumption leads to the, so-called, locally compact quantum groups, introduced and studied by Vaes and Kustermans [48], [49], Masuda, Nakagami and Woronowicz [57]. The theory of locally compact quantum groups is not as satisfying as for compact quantum groups since the existence of left and right invariant Haar measures has to be included in the axioms, differently from the case of compact quantum groups where this just follows.

In this thesis, however, we only deal with compact quantum groups. There are two main sources of examples of compact quantum groups: compact quantum groups corresponding to deformations of Lie algebras, defined by Drinfeld and Jimbo, and quantum symmetry groups, whose first examples were introduced by Wang [96], [97] and Van Daele [90]. In this thesis we will be interested in the latter. The two examples of quantum symmetry groups that will be discussed here are the free unitary and free orthogonal quantum groups. Their name is related to the fact that they share many structural properties with the reduced C^∗- and von Neumann algebras of free groups [4], [85]. They were introduced by Wang and Van Daele in [90] and thoroughly studied by Banica [4], [3]. On the probabilistic side, due to their combinatorial nature, the role played by the orthogonal and the unitary group in classical probability is analogous to the one played by the free orthogonal and free unitary quantum groups in noncommutative probability [7], [6], [17].

The main topic of the manuscript concerns probabilistic boundaries of random walks on dual discrete quantum groups. First of all, similarly to the classical case, from any compact quantum group it is possible to define a dual discrete quantum group. The correspondent (non unital) C^∗-algebra is given by the c₀-direct sum of matrix algebras labelled by finite dimensional representations of the compact quantum group. A comultiplication, dual to the multiplication on the compact quantum group, is uniquely defined and the existence of left and right invariant Haar weights is proved [88]. We remark that discrete quantum groups can be defined abstractly, as done in [88], and compact quantum groups can be characterised as duals of discrete quantum groups.

The theory of random walks on groups is an ever growing subject interconnected with multiple fields of study such as probability, harmonic analysis, potential theory, geometry and algebra, to state a few. Properties of random walks on groups, their asymptotic behaviour and their boundaries are related to algebraic characteristics of the group, like amenability, ex- ponential growth, and many others. Thus, also in the quantum picture, quantum random walks are interesting to analyse as they contain information on the relatedC^∗- and von Neu- mann algebras. The boundaries that will be examined here are a noncommutative version of the Poisson and Martin boundaries. For classical groups, the Poisson boundary is a measure space while the Martin boundary, introduced by Martin in [56], is a boundary in the topological sense. They both encode the asymptotic behaviour of random walks and were introduced to characterise harmonic functions. The Poisson boundary is related to bounded harmonic functions whereas via the Martin boundary it is possible to identify all not necessarily bounded, positive harmonic functions. Equipped with a certain measure, the Martin boundary is isomorphic to the Poisson boundary.

The study of probabilistic boundaries of discrete quantum groups was initiated by Biane

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who considered random walks on duals of compact Lie groups [11], [13], [12], and recovered a theory parallel to the one of random walks on discrete groups, for the latter see for example [100]. Working with von Neumann algebras of compact Lie groups, like SU(n), the sub- markovian operators considered were noncommutative analogues of convolution operators.

The Martin boundary of the dual ofSU(2)was computed by Biane and shown to be equal to the two-dimensional sphere.

Poisson boundaries for dual discrete quantum groups were defined by Izumi in [32]. They were introduced in connection with the study of infinite tensor product actions of quantum groups on factors. It turned out in fact that actions of quantum groups produced nontrivial relative commutant of the fixed point algebras differently from actions of genuine compact groups. The relative commutant of the fixed point algebras could be realised as Poisson boundaries of dual discrete quantum groups, with respect to special convolution Markov operators, having the property of preserving the centre of the von Neumann algebra. Cor- respondingly, a classical random walk can be defined. Studying general Markov operators does not bring interesting results, being too general and difficult to work with.

The classical Poisson boundary of the random walk on the centre naturally embeds in the noncommutative Poisson boundary, and by a result of Hayashi [30] it is trivial whenever the quantum group has commutative fusion rules, as it is forSU_q(2), computed by Izumi. It was shown by Izumi in [32], that the Poisson boundary ofSU_q(2)is given by the Podleś sphere.

The Poisson boundary for the dual ofSU_q(n)was computed by Izumi, Neshveyev and Tuset and shown to be equal to the quantum flag manifold [36]. This result was then extended by Tomatsu for the entire class of Drinfeld-Jimboq-deformations of Lie groups [81].

The Poisson boundary was also computed for the duals of compact quantum groups in the second class of examples introduced earlier: the free orthogonal quantum groups were examined by Vaes and Vergnioux in [85] while the free unitary quantum groups were studied by Vaes and Vander Vennet in [84]. The case of the free unitary quantum group is particularly interesting as the fusion rules are noncommutative, and the boundary can be viewed as a von Neumann algebra fibered over the classical Poisson boundary. Each fiber consists of an infinite tensor product of matrix algebras.

As one can deduce, we have a good understanding of the Poisson boundary of dual discrete quantum groups since it has been computed for the most important classes of examples.

The Martin boundary of dual discrete quantum groups was introduced by Neshveyev and Tuset in [63]. In general, the Martin boundary of dual discrete quantum groups is not well understood as it is for the Poisson boundary. The Martin boundary is substantially more difficult to compute, therefore still unknown for many classes of examples. The computation of the Martin boundary for the dual of SU_q(2)is due to Neshveyev and Tuset in [63] and shown to coincide with theC^∗-algebraic version of the Podleś sphere. There are partial results in [38] for the Martin boundary of SU_q(n). Concerning the class of free quantum groups, the Martin boundary of free orthogonal quantum groups was identified with aC^∗-algebraic version of higher dimensional Podleś spheres. Due to the few number of examples present in literature, the main objective of this thesis is to gain more insight into the Martin boundary theory for dual discrete quantum groups.

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Another question that remains open is whether the Poisson and Martin boundaries are connected, in some sense, as it happens for the classical case. It was conjectured by Neshveyev and Tuset in [62] that there exists an isomorphism between the Poisson boundary of a dual discrete quantum group and the enveloping von Neumann algebra of the Martin boundary, but currently there is no proof confirming this; it was shown in [39] that the conjecture holds forSU_q(2)but the problem remains unsolved in general.

Up until now we have only discussed the operator algebraic aspects of compact quantum groups but it appears that the language of categories is an important tool for the understanding of the noncommutative perspective. There are several duality theorems between (objects related to) quantum groups and categories having certain properties. To mention, the famous Woronowicz Tannaka-Krein duality [102], a quantum version of Tannaka-Krein duality for genuine compact groups. The theory of random walks was also extended to the categorical picture: the categorical Poisson boundary was defined by Neshveyev and Yamashita in [66]

and the categorical Martin boundary was later characterised by Jordans in [39].

In this thesis both the categorical and operator algebraic viewpoints play a fundamental role: results obtained in the analytical setting will be extended to the categorical one and in some cases the categorical approach will provide some simplifications in computations.

The thesis is structured in the following way: it consists of five chapters, two of which, the first and the third, contain preliminary material. In the others new results will be presented.

Below is a detailed description of the contents of each chapter.

1. Chapter 1 is dedicated to the discussion of the main concepts of compact and dual discrete quantum groups. It is possible to extend to the quantum picture many classical notions related to compact groups, but also new phenomena appear, making the theory more interesting. As mentioned previously, compact quantum groups can be studied from a purely categorical perspective in view of Woronowicz’s Tannaka-Krein duality theorem.

Thus, in this chapter we have a coexistence of analytical and categorical results.

2. Chapter 2, based on [53], contains an alternative proof of Woronowicz’s Tannaka-Krein duality theorem, as stated in [8] for the construction of new examples of free quantum groups. Given a finite dimensional Hilbert space and a collection of operators between its tensor powers satisfying certain properties, we give a short proof of the existence of a compact quantum group with a fundamental representationU such that the intertwiners between the tensor powers of U coincide with the given collection of operators. The general version of Woronowicz Tannaka-Krein duality is then deduced from this. Only basic tools of finite dimensional algebra are used.

3. Chapter 3 includes preliminary material on boundaries of dual discrete quantum groups.

The classical theory is also discussed in order to get a better understanding of the noncommutative perspective. Boundaries of random walks on categories are also introduced.

4. Chapter 4 is an extension of [54]. We here compute the Poisson and Martin boundaries for a special class of discrete quantum groups, that is, finite extensions of discrete quantum

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groups. It is shown that under suitable assumptions, the boundaries of a dual discrete quantum group coincide with the boundaries of its extension by a finite quantum group.

This includes the semi-direct product quantum group case [5], recently studied in [70].

This chapter also contains a discussion on traces on the von Neumann algebraic crossed product of a dual discrete quantum group by a finite group. The results are extended to the categorical framework.

5. In Chapter 5 the Martin boundary of duals of free unitary quantum groups is computed. It is the first example of Martin boundary of the dual of a non coamenable compact quantum group with noncommutative fusion rules. The Poisson boundary was obtained in [84] to- gether with aC^∗-algebraB_∞which was conjectured to be the Martin boundary. We prove that thisC^∗-algebra indeed coincides with the Martin boundary. The categorical perspective allows to compute the Martin boundary by separately understanding the asymptotic behaviour of perturbed random walks on binary trees. Random walks on trees have been extensively studied throughout the years by many authors. To mention, Cartier [15] who examined the nearest neighbourhood case, Dynkin and Malyutov [24] for the nearest neighbourhood homogeneous case, Derriennic [21] for the bounded range homogeneous case, and Picardello and Woess [69] for the more general bounded range not necessarily homogeneous case. Related to this study is also the work of Ancona in [1] who analysed random walks on hyperbolic spaces. Inspired by the above work we show that the perturbed random walks asymptotically behave like the classical random walks, recovering the whole Martin boundary of the dual of the free unitary quantum group.

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Chapter 1 Preliminaries

1.1 Quantum groups and their representations

In this chapter we build up the main theory that will be used throughout the thesis, introdu- cing compact/dual discrete quantum groups and their representation theory. It is divided into two parts: the first involves operator algebraic concepts, the second, categorical notions. The link between the two viewpoints is explained in Section 1.3. We essentially follow [64] and [63], but also refer to [80] and [46] for a more algebraic approach and to [45] for a categorical perspective. A relatively short and clear introduction is also given by [50].

1.1.1 Compact quantum groups

Before starting with the definitions we want to give some intuition. LetAbe a unital commut- ativeC^∗-algebra. By the well known theorem of Gelfand [59, Theorem 2.1.10.],Ais isomet- rically∗-isomorphic to the algebra C(X_A)of continuous functions on a compact Hausdorff spaceX_A. A unital∗-homomorphism between commutativeC^∗-algebrasAandBcorresponds to a continuous map between the compact spacesX_B andX_A. Thus, there is an equivalence between the category of commutative unitalC^∗-algebras and the category of compact topological spaces.

Consider now the commutativeC^∗-algebra of continuous functions on a compact groupG, C(G). Note thatC(G)⊗C(G) =C(G×G). We can define the following map

∆:C(G)→C(G)⊗C(G), such that ∆(f)(s,t) = f(s t),

for any s,t ∈G. Since the multiplication m onG is associative, f((s t)u) = f(s(t u)). This implies that the map ∆ has to satisfy the equality (∆⊗ι)∆ = (ι⊗∆)∆. By the cancel- lation property of the multiplication on G, functions of the form (s,t) → f₁(s)f₂(s t) for f₁,f₂∈C(G), separate points inG×G, thus, by the Stone-Weierstrass theorem, they span a dense subspace ofC(G×G). This can be rephrased by saying that∆(C(G))(C(G)⊗1)and

∆(C(G))(1⊗C(G))are dense inC(G)⊗C(G).

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It can be shown that any compact semigroup with cancellation is a group. This implies that the category of compact groups is equivalent to the category of unital commutativeC^∗- algebras equipped with a map∆satisfying the above properties.

The map ∆ can also be defined for more general unital, not necessarily commutative, C^∗-algebras, which, by the above considerations, can be thought of as algebras of continuous functions on a virtual (compact) geometric object, the so-called compact quantum group.

We now state the rigorous definition of a compact quantum group, due to Woronowicz.

Definition 1.1.1 (Woronowicz). A compact quantum group is a pair(C(G),∆) given by a unitalC^∗-algebraC(G)and a unital∗-homomorphism∆:C(G)→C(G)⊗C(G)such that

(i) (∆⊗ι)∆= (ι⊗∆)∆;

(ii) (C(G)⊗1)∆(C(G))and(1⊗C(G))∆(C(G))are dense inC(G)⊗C(G).

The first property is calledcoassociativityand the secondcancellation property.

We will sometimes refer toGas the compact quantum group, emphasising the underlying virtual object, but always mean the pair(C(G),∆).

Notation 1.1.2. All tensor products ofC^∗-algebras that appear, as in the definition above, are assumed to be minimal, unless stated otherwise.

We already examined the example(C(G),∆), whenC(G)is commutative, that is, when Gis a classical compact group. Another simple example of a compact quantum group is given by the following.

Example 1.1.3. Consider the reduced groupC^∗-algebra of a discrete groupΓ with canonical generators (λ_γ)_γ∈Γ. ThenC^∗_r(Γ)is a compact quantum group with comultiplication defined by∆(λ_γ) =λ_γ⊗λ_γ.Note that∆is invariant under the flip map, thus, it is said to becocom- mutative. This quantum group is usually denoted by(C(bΓ),∆).

Many concepts in the theory of compact groups can be generalised to compact quantum groups, keeping in mind that instead of working directly with the group (G,m) one has to translate notions to the dual picture(C(G),∆). One of these is the notion of the Haar measureµ_h, characterised by the property

Z

G

f(t s)dµ_h(t) = Z

G

f(t)dµ_h(t) = Z

G

f(s t)dµ_h(t),

for any continuous function f ∈C(G). In the dual pictureµ_h corresponds to a linear functional honC(G), which has to satisfy(h⊗ι)∆(f) =h(f)1= (ι⊗h)∆(f). This leads to the following theorem, [101, Proposition 4.1.]. LetGbe a compact quantum group, define the convolution of two functionalsω₁andω₂onC(G)byω₁∗ω₂= (ω₁⊗ω₂)∆.

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Theorem 1.1.4(Woronowicz). For any compact quantum groupG, there exists a unique state honC(G)which is left and right invariant, that is, such that

ω∗h=ω(1)h=h∗ω, for anyω∈C(G)^∗. The state his called Haar state.

Note that the Haar state is not faithful in general.

Example 1.1.5. LetΓ be a discrete group and consider the quantum group(C^∗_r(Γ),∆)defined in Example 1.1.3. The Haar state is the canonical trace onC^∗_r(Γ)which we know is faithful.

But the same is true if we consider the compact quantum group(C^∗(Γ),∆)given by the full groupC^∗-algebra with the same comultiplication, and it is a known fact that the canonical trace here is not faithful ifΓ is nonamenable.

The Haar state is not necessarily a trace as in the above example. It being a trace is related to properties of the quantum group and will be reviewed in the next sections.

Denote by π_h,L²(G),ξ_h

the GNS-triple corresponding to the Haar state, that is,π_h is the representation onL²(G)with cyclic vectorξ_h.

We now introduce the notion of a representation of a compact quantum group. Before we do so, let us first turn to classical compact groups and see what representations look like from the function algebra point of view.

Let G be a compact group. A representation ofG on a finite dimensional space H is a continuous homomorphism U ∈C(G,B(H)), such that for each s ∈G, U(s) is invertible.

IdentifyingC(G,B(H)) with B(H)⊗C(G), the function U can be viewed as an invertible element inB(H)⊗C(G). Then the homomorphism condition, U(s t) = U(s)U(t)can be written as(ι⊗∆)(U) =U₁₂U₁₃, whereU₁₂∈B(H)⊗C(G)⊗C(G)is equal to U⊗1andU₁₃ is equal to(ι⊗Σ)(U⊗1), denoting byΣ:C(G)⊗C(G)→C(G)⊗C(G)the map defined by Σ(a⊗b) =b⊗a, for anya,b∈C(G).

Definition 1.1.6. Arepresentation of a compact quantum groupG on a finite dimensional Hilbert spaceH, is an invertible element U∈B(H)⊗C(G)such that

(ι⊗∆)(U) =U₁₂U₁₃. It is unitary ifU is a unitary element inB(H)⊗C(G).

One either calls U a representation of G or a corepresentation of C(G). Analogously to the classical case, given two finite dimensional representations U ∈ B(H_U)⊗C(G) and V ∈B(H_V)⊗C(G)of a compact quantum groupG, we can define their direct sumU⊕V ∈ B(H_U ⊕H_V)⊗C(G)in the obvious way, and their tensor productU⊗V ∈B(H_U⊗H_V)⊗ C(G)byU⊗V =U₁₃V₂₃.

Given two vectorsξ andηin a Hilbert spaceH, denote bym_{ξ η}the matrix unit(·,η)ξ. If we fix an orthonormal basis (ξ_i)_i of H_U with matrix units(m_{i j})_{i j} and (η_l)_l of H_V with corresponding matrix units(n_{k l})_k,l, we can writeU=P

i,jm_{i j}⊗u_{i j} andV =P

k,ln_{k l}⊗v_{k l}.

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In terms of the entries ofU, the condition(ι⊗∆)(U) =U₁₂U₁₃becomes∆(u_{i j}) =P

ku_{i k}⊗ u_{k j}. The tensor product representationU⊗V is equal toP

i,j,k,lm_{i j}⊗n_{k l}⊗u_{i j}v_{k l}.

In the genuine compact group case, one can also take the dual or contragredient of a representation. That is, ifU is a representation of a compact groupGonH, the contragredient representation U^c is a representation on the dual space H^∗ defined by(U^c(g)ϕ)(ξ) = ϕ(U⁻¹(g)ξ)for anyϕ∈H^∗andξ ∈H.

An analogue of the contragredient representation can also be defined in the quantum setting. Consider a finite dimensional Hilbert space H and identify its dual H^∗ with the complex conjugate H¯, via the Riesz representation theorem. Let j:B(H) →B(H¯) be the

∗-anti-homomorphism whose square is equal to the identity, assigning to each operator its dual, namely j(T)ξ¯=T^∗ξ.

Definition 1.1.7. The contragredient representation of a finite dimensional representation U ∈B(H)⊗C(G)ofGis the element U^c ∈B(H¯)⊗C(G)defined by

U^c = (j⊗ι)(U⁻¹).

It is indeed a representation as it clearly satisfies (ι⊗∆)(U^c) =U₁₂^cU₁₃^c, and it can be shown to be invertible.

Note that if U is unitary this doesn’t imply that also U^c is, as for the classical case. As before, fixing an orthonormal basis(ξ_i)_i ofH with matrix units(m_{i j})_{i j}, denote by(ξ¯_i)_i the corresponding orthonormal basis ofH¯and by(m_{i j})_i,_j its matrix units. Then, ifU is unitary, the contragredient representation U^c is equal toP

i jm_{i j}⊗u_{i j}^∗.

Definition 1.1.8. The space ofintertwinersof two finite dimensional representationsU and V of a compact quantum groupG is given by

{T ∈B(H_U,H_V)|(T ⊗ι)U =V(T ⊗ι)}, and usually denoted byHom_G(U,V)orMor(U,V).

A representation U is called irreducibleifMor(U,U):=End(U)'C. Two representa- tionsUandV of a compact quantum groupGareequivalent, in which case we writeU'V, ifMor(U,V)contains an invertible element. Two irreducible representations U andV are either equivalent, in which caseMor(U,V)'C, orMor(U,V) =0. This is the content of Schur’s lemma translated in the quantum picture.

Remark 1.1.9. Every representation decomposes into a direct sum of irreducible representations and every irreducible representation is finite dimensional.

Remark 1.1.10. Note that for any two representations U,V of a compact quantum group G,U⊗V 'V ⊗U is not satisfied in general, as for compact groups.

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Definition 1.1.11. For every finite dimensional representation U of a compact quantum groupGthere exists a unique positive invertible operatorρ_U ∈Mor(U,U^{c c})⊆B(H_U)satisfying

Tr(Tρ_U) =Tr(Tρ⁻¹_U ), for anyT ∈End(U).

The numberdim_qU=Tr(ρ_U)is calledquantum dimensionofU.

By Cauchy-Schwarz inequality, dimH_U ≤dim_qU. Equality holds if and only ifρ_U =1. As previously mentioned, ifU is unitary this doesn’t imply thatU^c is, but as in the classical case, every representation is equivalent to a unitary one.

Definition 1.1.12. Theconjugaterepresentation of U is the unitary representationU¯ equivalent to U^c defined by

U¯ = (j(ρ_U)^1/2⊗ι)U^c(j(ρ_U)^−1/2⊗ι).

Given a compact quantum group G, let Irr(G) be the set of equivalence classes of irreducible unitary representations of G. For any s ∈ Irr(G), we fix representatives U_s ∈ B(H_s)⊗C(G)and denote simply byρ_s=ρ_U_s ∈B(H_s)the unique positive invertible operator satisfyingρ_s ∈Mor(U_s,U_s^{c c})andTr(ρ_s) =Tr(ρ⁻¹_s ):=dim_q(s). Denote by¯s the element in Irr(G)corresponding to the conjugate representation ofU_s,U¯_s∈B(H¯_s)⊗C(G). The tensor product of two representationsU^s andU^t decomposes into a direct sum of irreducible representations,U_s⊗U_t 'L

r∈Irr(G) N_s,t^r U_r, whereN_s,t^r is the multiplicity ofU_r inU_s⊗U_t, that is,N_s,t^r =dim Mor(U_r,U_s⊗U_t). Thus, we will sometimes writes⊗t'L

r∈Irr(G) N_s,t^r r. We introduce two families(ψ_s)_s and(φ_s)_sof positive linear functionals which will play a fundamental role in the theory of random walks on quantum groups treated in the following chapters. Defineψsandφs inB(H_s)^∗by

ψ_s(T) =dim_q(s)⁻¹Tr(ρ_sT), φ_s(T) =dim_q(s)⁻¹Tr(ρ⁻¹_s T). (1.1.1) By assumption,ψ_s andφ_s agree onEnd(U_s). The following equalities hold:

ψ_s(T)1_s= (ι⊗h)(U_s(T ⊗1)U_s^∗), φ_s(T)1_s= (ι⊗h)(U_s^∗(T ⊗1)U_s). (1.1.2) Choosing T equal to the matrix unit m_{l j}^s ∈ B(H_s), we obtain the orthogonality relations [101],

h(u_{k l}^s u_{i j}^t^∗) = δ_kiδ_{s t} (ρ_s)_{j l}

dim_q(s), h(u_{i j}^t^∗u_{k l}^s ) = δ_{j l}δ_{s t} (ρ⁻¹_s )_ki

dim_q(s). (1.1.3) The linear span of matrix coefficients of all finite dimensional representations ofGis denoted by

C[G] = span{(ω⊗ι)(U_s)|for anyω∈B(H_s)^∗ands ∈Irr(G)},

and is a Hopf∗-algebra. That is, fixing orthonormal bases of H_s for any s ∈Irr(G), we can define thecounit"and theantipodeS by"(u_{i j}^s) =δ_{i j} andS(u_{i j}^s ) =u_{j i}^s^∗, for1≤i,j ≤dimH_s.

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It is enough to define the counit and the antipode on such elements, since by (1.1.3) they form a linear basis inC[G].

Remark 1.1.13. Note that"andS don’t necessarily extend to the wholeC^∗-algebraC(G).

Definition 1.1.14. A compact quantum groupG is ofKac typeif one of the following equivalent conditions holds:

(i) the antipode S extends continuously to a∗-anti-homomorphismS:C(G)→C(G);

(ii) S²=ι;

(iii) S is∗-preserving;

(iv) the Haar stateh:C(G)→Cis a trace.

We now want to introduce theright regular representationof a compact quantum group. It will allow us to develop the Peter-Weyl theory for compact quantum groups and to implement the duality between compact quantum groups and dual discrete quantum groups, which we define in the next section.

In the classical compact group case, the right regular representationW is defined by (W_sf)(t) = f(t s), for f ∈L²(G). (1.1.4) We can view W as an element of the multiplier algebra M(K(L²(G))⊗C(G)) which we consider acting on L²(G)⊗L²(G). Then, for f₁ ∈C(G), f₂ ∈ L²(G) and for any s,t ∈ G, (1.1.4) translates to

(W(f₁⊗f₂))(t,s) = (W_sf₁)(t)f₂(s) = f₁(t s)f₂(s) = (∆(f₁)(1⊗f₂))(t,s).

Assume now thatG is a compact quantum group and thatC(G)acts faithfully and non degenerately on a Hilbert spaceH₀. Recall thatL²(G)denotes the Hilbert space of the GNS- construction with respect to the Haar state and write simplyΛ_h(a)∈L²(G)forπ_h(a)ξ_h. Let W be the element inM(K(L²(G))⊗C(G))defined by

W(Λh(a)⊗η) = (πh⊗ι)∆(a)(ξh⊗η), for alla∈C(G) andη∈H₀. (1.1.5) ThenW is a unitary element and satisfies (ι⊗∆)(W) =W₁₂W₁₃. Hence, it is an infinite dimensional representation ofG(unless G is finite) called theright regular representationof G. It can be shown that

(i) the space{(ω⊗ι)(W)|ω∈K(L²(G))^∗}is dense inC(G);

(ii) W(π_h(a)⊗1)W^∗= (π_h⊗ι)∆(a)(ξ_h⊗1)for alla∈C(G).

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That is, W implements the comultiplication ∆. As a corollary, the linear span C[G] of matrix coefficients of finite dimensional irreducible representations of G is dense in C(G).

Moreover, using (ii) and the fact thatW is a representation we see thatW satisfies thepentagon equationW₂₃W₁₂=W₁₂W₁₃W₂₃.

For any ξ and η in a Hilbert space H denote by ω_ξ,η ∈ B(H)^∗ the linear functional defined byω_ξ_,_η(T) = (Tξ,η). The right regular representation decomposes into irreducibles as follows. For each s ∈Irr(G)choose an orthonormal basis for H_s, say(ξ_i^s)_i. For each ξ_i^s one can check directly that the map

T_i^s:H_s→L²(G), such that T_i^sξ_j^s=dim_q(s)¹^/²(ω_ξs

j,ρ¹s^/²ξ_i^s⊗ι)(U_s)ξh,

intertwinesU^sandW. Moreover, by the orthogonality relations (1.1.3), fori 6=j the images of the mapsT_i^sandT_j^swill be orthogonal inL²(G). From previous considerations, the images of the maps (T_i^s)_i for any s ∈Irr(G) span a dense subspace of L²(G). Thus, we obtain a decomposition into irreducibles forW where each U_s appears with multiplicitydimH_s, so we write

W = Y

s∈Irr(G)

U_s. (1.1.6)

Any compact quantum group has a reduced form and a universal one depending on how C[G] is completed to a C^∗-algebra. As the word suggests, the reduced version of G, denoted by (C(G_r),∆_r), is given by the completion of π_h(C[G]) ⊂ B(L²(G)) with respect to the GNS-representation corresponding to the Haar state. The Haar state on C(G_r) is known to be faithful. Since h = (h⊗h)∆, this implies that the coproduct ∆ determines a coproduct ∆_r:C(G_r) → C(G_r)⊗C(G_r) satisfying ∆_rπ_h = (π_h⊗π_h)∆. The unitary W_r = (ι⊗πh)(W) ∈B(L²(G)⊗L²(G)) is called themultiplicative unitary, as in the more general duality theory in operator algebras described in [2]. The comultiplication ∆_r is implemented by W_r, ∆_r(a) = W_r(a ⊗1)W_r^∗. Furthermore, ∆_r extends to a normal ∗- homomorphism L^∞(G) → L^∞(G)⊗¯L^∞(G), where L^∞(G) denotes the von Neumann al- gebraπ_h(C(G))⁰⁰⊆B(L²(G)), and is called thevon Neumann algebraofG.

Theuniversal versionofG,(C(G_u),∆_u), is instead given by the completion ofC[G]with respect to the universal norm defined by

kak_u =sup{kπ(a)ks.t.π:C[G]→B(H) is a unital∗-homomorphism}.

By universality the comultiplication∆:C[G]→C[G]⊗C[G]extends to a homomorphism

∆u:C(G_u)→C(G_u)⊗C(G_u), satisfying the same properties.

A compact quantum group C(G) always sits between its reduced and universal version, that is, there are surjective homomorphismsC(G_u)→C(G)→C(G_r).

Definition 1.1.15. G iscoamenableif the above morphisms are isomorphisms. Or equivalently, if the counit"extends to a character ofC(G_r), [9].

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1.1.2 Dual discrete quantum groups

The goal of this subsection is to define the dual discrete quantum group. Consider the space C[G]^∗=:U(G) of linear functionals on C[G]. It is a unital∗-algebra with multiplication given by the convolutionω₁∗ω₂= (ω₁⊗ω₂)∆, involutionω^∗=ω¯S, whereω(¯ a) =ω(a^∗), and unit". Dual to the multiplication onC(G), the comultiplication onU(G)can be defined by ∆(ω)(aˆ ⊗b) =ω(ab), for a,b∈C[G].

Note that ∆ˆ is a unital ∗-homomorphism U(G) → (C[G]⊗C[G])^∗ =: U(G×G), and, unlessC[G]is finite dimensional, its image is not contained in the algebraic tensor product U(G)⊗U(G). By duality, one can define onU(G)the antipodeS(ω) =ˆ ωS and the counit

"(ω) =ˆ ω(1). Any finite dimensional unitary representation U of G gives rise to a unital

∗-representation

π_U:U(G)→B(H_U), by π_U(ω) = (ι⊗ω)(U). (1.1.7) For any s ∈ Irr(G) let π_s := π_U_s. Through the collection (π_s)_s∈Irr₍_G₎ one gets the iso- morphismU(G)'Q

s∈Irr(G)B(H_s).In this picture, the comultiplication∆(ω)ˆ of an element ω∈ U(G)is defined as the unique element such that

(π_s⊗π_t)∆(ω)Tˆ =Tπ_r(ω), for all T ∈Mor(U_r,U_s⊗U_t). (1.1.8) Since(U(G), ˆ∆)is not a Hopf∗-algebra in general, we restrict ourselves to a∗-subalgebra of U(G)which we denote byC[Gˆ], defined by

C[G] =ˆ {aˆ:=h(·a)|a∈C[G]}.

Analogously to the classical case, the bijection F:C[G]→C[G],ˆ a → F(a) =a, is calledˆ Fourier transform [89] and gives an isomorphism of linear spaces. Note that C[Gˆ]has no identity since"does not belong toC[G]. In order to define the dual discrete quantum groupˆ we have to construct a C^∗-algebra fromC[G]. Define for each elementˆ ϕ ∈C[G], the op-ˆ eratorπ(ϕ)ˆ acting on the dense subspace of L²(G)byπ(ϕ)ˆ Λ_h(a) = (ι⊗ϕ)∆(a)ξ_h ∈L²(G).

Thenπ(ϕ)ˆ extends to a bounded operator onL²(G)andπˆis a faithful∗-representation. Note thatπˆis exactlyπ_W, as defined in (1.1.7), whereW is the right regular representation (1.1.5).

Denote byc₀(Gˆ)the completion ofπ(C[ˆ Gˆ])⊆B(L²(G)).

By looking at the Fourier transform of coefficients of irreducible representations of G, using the orthogonal relations (1.1.3) and the representations (πs)_s∈Irr₍G), we can identify C[G]ˆ with the algebraic direct sum L

s∈Irr(G)B(H_s). Then, using the same identification, c₀(G)ˆ is given by c₀-L

s∈Irr(G)B(H_s) and `^∞(G), equal to the multiplier algebraˆ M(c₀(G)),ˆ by `^∞-L

s∈Irr(G)B(H_s). Denote by I_s the identity in B(H_s), so (I_s)_s∈Irr₍_G₎ are the minimal projections in the centre of c₀(G). We define the dual discrete quantum group to be theˆ virtual object(G, ˆˆ ∆)and refer either to the pair(c₀(Gˆ), ˆ∆)or to(`^∞(Gˆ), ˆ∆).

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To summarise, U(G) ' Y

s∈Irr(G)

B(H_s) ={(x_s)_s s.t. x_s∈B(H_s)}, C[G]ˆ ' M

s∈Irr(G)

B(H_s) ={(x_s)_s∈Y

s

B(H_s) s.t. x_s=0 for all but finitely many s}, c₀(Gˆ) ' c₀- M

s∈Irr(G)

B(H_s) ={(x_s)_s∈Y

s

B(H_s) s.t. (kx_sk)_s∈c_o(Irr(G))},

`^∞(Gˆ) '`^∞- M

s∈Irr(G)

B(H_s) ={(x_s)_s∈Y

s

B(H_s) s.t. (kx_sk)_s∈`^∞(Irr(G))}.

(1.1.9) The comultiplication ∆ˆ:`^∞(Gˆ) → `^∞(Gˆ)⊗`¯ ^∞(Gˆ) is defined by (1.1.8), interpreting the representationπ_s as the projectionQ

s∈Irr(G)B(H_s)→B(H_s)onto the blockB(H_s). Since the right regular representationW belongs toM(c₀(Gˆ)⊗C(G)), identity (1.1.8) implies that(∆⊗ˆ ι)(W) =W₁₃W₂₃, which again, using the pentagon equation, shows that∆ˆ is implemented byW, that is,∆(x) =ˆ W^∗(1⊗x)W.

Discrete quantum groups can also be defined abstractly, as done in [88]. They are quantum groups associated to aC^∗-algebra that is thec₀-direct sum of matrix algebras. Then compact quantum groups are defined as duals of discrete quantum groups.

Denote byρtheWoronowicz character, that is, the unique unbounded element inU(G) satisfyingπ_s(ρ) =ρ_sfor anys ∈Irr(G). It is group-like, meaning that it satisfies∆(ρ) =ˆ ρ⊗ρ;

furthermore,Sˆ(ρ) =ρ⁻¹.

For dual discrete quantum groups it is no longer possible to define a left and right invariant Haar state in general. But there exist unique left invariant and right invariant weights

ˆh_L= X

s∈Irr(G)

dim_q(s)²ψ_s, ˆh_R(·) = X

s∈Irr(G)

dim_q(s)²φ_s,

where, recall,ψ_s andφ_s were defined in (1.1.1). Their modular groups areσ_t^L=Adρ^{i t} and σ_t^R=Adρ^{−i t} respectively. It is possible to define two inner products onC[G], byˆ

(x,y)_L=ˆh_L(xσ₋^Li 2

(y^∗)), (x,y)_R=ˆh_R(xσ₋^Ri 2

(y^∗)). (1.1.10) If the Woronowicz characterρis equal to1, the compact quantum groupGis of Kac type.

Thus, this condition should be added to the equivalent ones appearing in Definition 1.1.14.

1.1.3 Actions of compact and discrete quantum groups

In this section we discuss actions of compact or discrete quantum groups onC^∗or von Neu- mann algebras. We also refer to [18] for a complete exposition.

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Assume thatG is a compact group. A continuous left action ofG on aC^∗-algebraB is a continuous mapG×B→B,(g,b)7→α_g(b), such that (i)α_g◦α_h=α_{g h}and (ii) for any g∈G each αg is a∗-automorphism. From the function algebra point of view this is equivalent to saying that there is a mapα:B →B⊗C(G)'C(G,B),b 7→ {α(b): g →α_g(b)}, satisfying (α⊗ι)α = (ι⊗∆)α and that α(B)(1_B⊗C(G)) is dense in B ⊗C(G). The first condition corresponds to (i) and the second to (ii). With this in mind we introduce the following definition.

Definition 1.1.16. Aright actionof a compact quantum groupGon a unitalC^∗-algebraB is a∗-homomorphism

α:B →B⊗C(G), such that (α⊗ι)α= (ι⊗∆)α,

and such thatα(B)(1_B⊗C(G))is dense inB⊗C(G). One equivalently says thatα is a right coactionofC(G)on theC^∗-algebraB. Similarly, aleft actionofGonB is a∗-homomorphism α:B →C(G)⊗B such that(ι⊗α)α= (∆⊗ι)α and such thatα(B)(C(G)⊗1_B)is dense in C(G)⊗B. A unitalC^∗-algebraB equipped with an action of a compact quantum groupG is called aG-C^∗-algebra.

Note that any finite dimensional representationU ofGdefines maps δ_U^r:H_U →H_U⊗C[G], ξ 7→U(ξ ⊗1), δ_U^l :H_U →C[G]⊗H_U, ξ 7→U₂₁^∗(1⊗ξ), which are right and leftC[G]-comodule maps respectively, that is,

(ι⊗∆)δ_U^r = (δ_U^r ⊗ι)δ_U^r, (ι⊗")δ_U^r =ι,

and similar equalities hold forδ_U^l . TheC[G]-comodule map δ_U^r is calledunitaryif

〈δ_U^r(ξ),δ_U^r(η)〉= (ξ,η)1, for allξ,η∈H_U,

where 〈 ·,· 〉 is defined by〈ξ ⊗a,η⊗b〉= (ξ,η)a^∗b ∈C[G], and in such case the Hilbert spaceH_U is called a finite dimensional unitaryC[G]-comodule. Similarly forδ_U^l .

Vice versa, to any unitaryC[G]-comodule map from a finite dimensional Hilbert spaceH corresponds a unitary finite dimensional representationUofGontoH, see [18, Lemma 1.7.].

The above duality remains true even if the Hilbert space H is not necessarily finite dimensional. Correspondingly we will have a unitary infinite dimensional representation in the multiplier algebraM(K(H)⊗C(G)).

Assume α is a left action of G on a unital C^∗-algebra B. For s ∈ Irr(G), denote by Mor(s,α)the left comodule maps intertwiningδ_s^l andα, that is,

Mor(s,α) ={T:H_s→B|α(Tξ) = (ι_G⊗T)δ_s^l(ξ), ∀ξ ∈H_s}.

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Thespectral subspacecorresponding to s, is given by

Bs =span{Tξ |ξ ∈H_s and T ∈Mor(s,α)} ⊂B.

Consider the∗-subalgebraB ⊂Bconsisting of the linear span of spectral subspaces(B_s)_s∈Irr₍_G₎. Via the orthogonality relations one can check that

B=span{(h⊗ι)((a⊗1)α(b)) | a∈C[G], b∈B}, (1.1.11) where h, recall, is the Haar state on C(G), and that B is indeed dense in B. It is usually called theregular subalgebraofB, and its elements are calledregular.

Thefixed point algebraof a left actionαof a compact quantum group Gon aC^∗-algebra B is theC^∗-algebra

B^α={b∈B|α(b) =1⊗b} ⊆B.

There is a canonical conditional expectation

E_α:B→B^α, defined by E_α(b) = (h⊗ι)α(b).

It can be checked directly, using invariance of the Haar state, that E_α(b) indeed belongs to B^α. The action α is ergodic ifB^α 'C. An invariant statefor α is a stateϕ on B such that (ι⊗ϕ)α=ϕ(·)1. If the actionα is ergodic, there is a unique invariant stateϕ onB defined byϕ(b)1=E_α(b).

Definition 1.1.16 makes sense also ifG is replaced by a discrete quantum groupGˆ. Definition 1.1.17. A right action of a discrete quantum group Gˆ on a unital C^∗-algebra B is a ∗-homomorphism β:B → M(B⊗c₀(Gˆ)) satisfying (β⊗ι)β = (ι⊗∆)β, and thatˆ β(B)(c₀(Gˆ)⊗ 1_B)is dense inc₀(Gˆ)⊗B.

Definition 1.1.18. Aright action of a compact (resp.discrete) quantum groupG (resp. Gˆ) on a von Neumann algebraM is an injective, normal, unital∗-homomorphism

β:M →M⊗¯L^∞(G), such that (β⊗ι)β= (ι⊗∆)β, (resp.β:M→M⊗`¯ ^∞(G), such thatˆ (β⊗ι)β= (ι⊗∆)β).ˆ

The density condition does not have to be included in the definition as it is automatic for von Neumann algebras by [82].

Definition 1.1.19. AssumeB is a unitalC^∗-algebra equipped both with a left actionα:B→ C(G)⊗B of a compact quantum groupG and a right actionβ:B →M(B⊗c₀(Gˆ)) of the dual discrete quantum groupG. Thenˆ βdefines a left action ofC[G]onB,

.:C[G]⊗B →B, by x.a= (ι⊗x)β(a),

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such that x.a^∗ = (S(x)^∗.a)^∗. The triple (B,α,β) is a Yetter–DrinfeldG-C^∗-algebra if the compatibility condition

α(x.a) =x₍₁₎a₍₁₎S(x₍₃₎)⊗(x₍₂₎.a₍₂₎), (1.1.12) holds for all x∈C[G]anda∈ B.

In the above definition we used Sweedler’s notation, i.e., ∆(x) = x₍₁₎⊗x₍₂₎ and α(a) = a₍₁₎⊗a₍₂₎.

Definition 1.1.20. A Yetter-DrinfeldG-C^∗-algebra(B,α,β)is said to bebraided-commutative if

ab=b₍₂₎(S⁻¹(b₍₁₎).a), for anya,b ∈ B. (1.1.13) We now introduce adjoint actions of a compact quantum group on a finite dimensional algebra. Recall the families of functionals(ψ_s)_s and(φ_s)_s defined in (1.1.1).

Lemma 1.1.21(Izumi [32]). LetU∈B(H_U)⊗C(G)be a finite dimensional unitary representation of a compact quantum groupG. Then

(i) the∗-homomorphismα_U:B(H_U)→B(H_U)⊗C(G), defined byα_U(x) =U(x⊗1)U^∗for anyx ∈B(H_U), is a right action ofGonB(H_U).

(ii) The linear functional ψ_s =dim_q(s)⁻¹Tr(π_s(ρ)·)∈B(H_s)^∗ is the unique invariant state for the actionα_s(·) =U_s(· ⊗1)U_s^∗.

Symmetrically, lettingU₂₁∈C(G)⊗B(H_U),

(i⁰) the∗-homomorphismβ_U:B(H_U)→C(G)⊗B(H_U)defined byβ_U(x) =U₂₁^∗(1⊗x)U₂₁ for anyx∈B(H_U), is a left action ofGonB(H_U).

(ii⁰) The linear functionalφ_s =dim_q(s)⁻¹Tr(π_s(ρ⁻¹)·)∈B(H_s)^∗is the unique invariant state for the actionβ_s(·) = (U_s)^∗₂₁(1⊗ ·)(U_s)₂₁.

Moreover, the fixed point algebrasB(H_U)^α^U andB(H_U)^β^U are equal and coincide with the com- mutantπ_U(U(G))⁰, forπ_U defined in(1.1.7).

It is easy to check thatα_U andβ_U define right/left actions ofGonB(H_U). Conditions (ii) and (ii⁰) follow from the equalities in (1.1.2) and previous considerations on the connection between invariant states of ergodic actions and the canonical conditional expectations.

Notation 1.1.22. We will denote by C_α ⊂ c₀(Gˆ)^∗ the norm closure of the linear span of functionals (ψ_s)_s and analogously by C_β ⊂ c₀(G)ˆ ^∗ the norm closure of the linear span of (φ_s)_s. That is, elementsψ_µ∈ C_α andφ_µ∈ C_βwill be of the form

ψ_µ= X

s∈Irr(G)

µ(s)ψ_s, φ_µ= X

s∈Irr(G)

µ(s)φ_s, (1.1.14)

Poisson and Martin boundaries of discrete quantum groups: a noncommutative and categorical perspective