On the Reduced Operator Algebras of Free Quantum Groups

Transcription

1 On the Reduced Operator Algebras of Free Quantum Groups by Michael Paul Brannan A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy Queen s University Kingston, Ontario, Canada July 2012 Copyright c Michael Paul Brannan, 2012

2 Abstract In this thesis, we study the operator algebraic structure of various classes of unimodular free quantum groups, including the free orthogonal quantum groups O n +, free unitary quantum groups U n +, and trace-preserving quantum automorphism groups associated to finite dimensional C -algebras. The first objective of this thesis to establish certain approximation properties for the reduced operator algebras associated to the quantum groups G = O + n and U + n, (n 2). Here we prove that the reduced von Neumann algebras L (G) have the Haagerup approximation property, the reduced C -algebras C r (G) have Grothendieck s metric approximation property, and that the quantum convolution algebras L 1 (G) admit multiplier-bounded approximate identities. We then go on to study trace-preserving quantum automorphism groups G of finite dimensional C -algebras (B, ψ), where ψ is the canonical trace on B induced by the regular representation of B. Here, we extend several known results for free orthogonal and free unitary quantum groups to the setting of quantum automorphism groups. We prove that the discrete dual quantum groups Ĝ have the property of rapid decay, the von Neumann algebras L (G) have the Haagerup approximation property, and that L (G) is (in most cases) a full type II 1 -factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and i

3 uniqueness of trace for the reduced C -algebras C r (G), and the existence of multiplierbounded approximate identities for the convolution algebras L 1 (G). We also show that when B is a full matrix algebra, L (G) is an index 2 subfactor of L (O n + ), and thus solid and prime. Finally, we investigate strong Haagerup inequalities in the context of quantum symmetries arising from actions of free quantum groups on non-commutative random variables. We prove a generalization of the strong Haagerup inequality for -free R- diagonal families due to Kemp and Speicher, and apply this result to study strong Haagerup inequalites for the free unitary quantum groups. ii

4 Acknowledgments First and foremost, I thank my thesis supervisors James A. Mingo and Roland Speicher for all of their support and guidance. In particular, I thank them for introducing me to the beautiful world of free quantum groups (which ended up being the central theme of this thesis), and for the many fruitful discussions and encouragement over the years. I would also like to thank my master s thesis supervisors Brian Forrest and Nico Spronk at the University of Waterloo, who taught me a great deal of mathematics. The knowledge and experience I gained from Brian and Nico has influenced my doctoral work in a very positive way. I am also very grateful for their continued support and encouragement, even after completing my master s degree. I thank the members of my examination committee (Todd Kemp, Abdol-Reza Mansouri, Kayll Lake and Ram Murty) for giving up their time to read through this work. I also thank Teodor Banica, Benoît Collins, Matt Daws, Amaury Freslon, Ronald Kerman, Ebrahim Samei, Georges Skandalis, Roland Vergnioux, Christian Voigt and Moritz Weber for all of their support and advice. Finally, I thank my family for their constant love and support. In particular, I thank my wife Janina for always encouraging me, putting up with my occasional mathematics-induced grumpiness, and for making these last four years so enjoyable. iii

5 To My Wife Janina iv

6 Contents Abstract Acknowledgments Contents i iii v Chapter 1: Introduction Motivation Thesis Organization and Overview of Main Results Chapter 2: Background and Notation Operator Algebras Tensor Products Completely Bounded Maps The GNS Representation and L 2 -Extensions of CP Maps Nuclearity, Injectivity, and II 1 -Factors Compact and Discrete Quantum Groups Basic Examples Representation Theory Orthogonality Relations Universal Quantum Groups Discrete Quantum Groups Noncommutative Probability Spaces and Free Independence Non-Crossing Partitions and Free Cumulants R-Diagonal Elements and Free Complexification Chapter 3: Approximation Properties for Free Orthogonal and Free Unitary Quantum Groups Chapter Overview Introduction The Haagerup Property for Compact Quantum Groups of Kac Type The Haagerup Property for O n + and U n v

7 3.4.1 The Orthogonal Case The Unitary Case The Reduced C -Algebras and the Metric Approximation Property The Metric Approximation Property for C r (O n + ) The Metric Approximation Property for C r (U n + ) Chapter 4: Structure Results for Trace-Preserving Quantum Automorphism Groups Chapter Overview Introduction Quantum Automorphism Groups of Finite Dimensional C -algebras Representation Theory and the 2-cabled Temperley-Lieb Category Explicit Models Approximation Properties and The Property of Rapid Decay The Haagerup Property The Property of Rapid Decay Further Approximation Properties A Remark on Exactness Algebraic Structure Factoriality and Fullness Analysis of the operator T Simplicity of C r (G) The Structure of L (G aut (M n (C), tr)) Applications Proof of Theorem Chapter 5: Quantum Symmetries and Strong Haagerup Inequalities Chapter Overview Introduction Preliminaries and Notation Quantum Groups and Invariant Distributions The Hyperoctahedral Quantum Group Strong Haagerup Inequalities Proof of Theorem Properties of NC ɛ d (2dm) and Estimates of (A), (B), and (C) Strong Haagerup Inequalities for Bi-Invariant Arrays Application to the Metric Approximation Property Proof of Theorem Applications to Free Unitary Quantum Groups vi

8 Chapter 6: Concluding Remarks Future Work Bibliography 198 vii

9 1 Chapter 1 Introduction 1.1 Motivation Historically, there has always been a close connection between the theory of operator algebras (i.e., algebras of bounded linear operators on Hilbert spaces) and the harmonic analysis and representation theory of groups. This is largely due to the fact that unitary representations of groups generate interesting and highly non-trivial examples of operator algebras. In this thesis, we study an analogous connection that exists between the theory of operator algebras and certain mathematical objects called quantum groups. Roughly speaking, a quantum group is a generalization of the notion of a group within the framework of non-commutative geometry. 1 We will postpone the formal definition of a quantum group to Chapter 2, and in this chapter just content ourselves with some motivating remarks. In order to put into context and set the stage for the 1 The term quantum group was introduced by V. Drinfeld in [41]. The usage of the word quantum or quantization in the quantum group literature refers to mathematical concept of deforming the commutative algebra of coordinate functions on the phase space of a physical system, to obtain a new algebra of non-commutative coordinates. The mathematical notion of deformation was originally inspired by theory of quantum integrable models and exactly solvable models of statistical physics. See [45] for a good account of the history and origins of quantum groups.

10 1.1. MOTIVATION 2 quantum groups and operator algebras that we study, let us start from the beginning and first recall how, from a group, we can obtain interesting examples of operator algebras. Consider a discrete group Γ. Associated to Γ we have the complex group algebra C[Γ] (of finitely supported functions f : Γ C) equipped with its usual convolution product and involution f (γ) = f(γ 1 ), (γ Γ). The algebra C[Γ] admits a natural faithful tracial state τ : C[Γ] C, given by τ(f) = f(e) for each f C[Γ] (where e denotes the unit element of Γ). Using τ, we can define a non-degenerate inner product on C[Γ], given by f 1 f 2 = τ(f1 f 2 ). The completion of C[Γ] with respect to this inner product yields the Hilbert space l 2 (Γ) of square summable functions ξ : Γ C. Now consider the action C[Γ] on itself by left multiplication: (f, ξ) f ξ. This action is easily seen to extend to an injective -representation λ : C[Γ] B(l 2 (Γ)), called the left regular representation. Thus, starting from any discrete group Γ, we get a self-adjoint algebra λ(c[γ]) of bounded linear operators on the Hilbert space l 2 (Γ). If we now take certain topological closures of λ(c[γ]) in B(l 2 (Γ)), we obtain two very important examples of operator algebras: the reduced group C -algebra C r (Γ) = λ(c[γ]) B(l 2 (Γ)) B(l 2 (Γ)), and the group von Neumann algebra V N(Γ) = λ(c[γ]) WOT = λ(c[γ]) B(l 2 (Γ)), where WOT denotes the weak operator topology on B(l 2 (Γ)). Since the very beginnings of the theory of operator algebras in the early 20th

11 1.1. MOTIVATION 3 century, the algebras Cr (Γ) and V N(Γ) constructed from discrete groups Γ have consistently provided a bounty of examples which both enrich and motivate the general theory. To give just a few examples demonstrating this fact, we mention Popa s very modern deformation/rigidity theory for von Neumann algebras [75, 78, 79, 80], the theory of approximation properties for groups initiated by Haagerup in his seminal work [52], and Voiculescu s free probability theory [72]. Of particular relevance to this thesis is free probability theory, which is a modern branch of functional analysis invented in the 1980 s by D. V. Voiculescu, as an attempt to shed light on the structure of the mysterious free group factors V N(F k ). (F k is the free group on k 2 generators). A central question concerning these algebras is the (still unresolved) isomorphism problem: Does an isomorphism V N(F k ) = V N(F k ) imply that k = k? The novel idea of Voiculescu was to view V N(F k ) as an algebra of non-commutative random variables equipped with a distinguished expectation functional τ (given by the unique extension to V N(F k ) of the canonical trace τ : C[F k ] C). From this perspective, we may regard the generators {λ(g 1 ),..., λ(g k )} V N(F k ) (corresponding to the standard generators g i of the ith copy of Z in the free product F k = k i=1z) as random variables which satisfy a certain highly non-commutative form of probabilistic independence relative to the expectation τ which we call free independence 2 Over the last three decades, many deep parallels have emerged between classical and free probability theory. For instance, there exist free analogues of cumulants, 2 To further justify why it is reasonable to think of the above situation as an analogy with classical independence, observe that if we replace F k with the rank k free abelian group Z k, then the Fourier transform yields an isomorphism between V N(Z k ) and the algebra of random variables L (T k, dg) (where dg = Haar probability measure on T k ). Moreover, this isomorphism identifies τ : V N(Z k ) C with the expectation f T k fdg, and it is easy to verify that the standard generators {λ(g 1 ),..., λ(g k )} V N(Z k ) are stochastically independent with respect to τ.

12 1.1. MOTIVATION 4 Brownian motion, entropy and the central limit theorem (where the universal role of the Gaussian law is played by Wigner s semicircle law from statistical mechanics). These notions and analogies have led to striking advancements in our understanding of the free group factors, and more generally, free products of operator algebras. In addition, this theory has produced several connections to other fields, including random matrices, combinatorics, and even electrical engineering (see [72, 88], for example). In this thesis, we study operator algebras that simultaneously generalize the algebras Cr (Γ) and V N(Γ) coming from discrete (in particular, free) groups Γ, and the function algebras C(G) and L (G) associated to compact groups G. The objects of interest in this thesis that give rise to such operator algebras are called free quantum groups. To motivate the notion of a free quantum group from a probabilistic perspective, consider an n-tuple X = {x i } n i=1 L (Ω, P ) of real, independent and identically distributed Gaussian random variables, viewed as a random vector in R n. (Here, L (Ω, P ) is the algebra of complex random variables with finite moments of all orders.) It is well known that the joint distribution of X is rotationally invariant: for any orthogonal matrix U O n, the rotated random vector UX has the same distribution as X. In other words, if we let {u ij } 1 i,j n C(O n ) denote the standard coordinate functions on O n and let x τ(x) = x(ω)dp (ω) denote the expectation Ω functional, then the C(O n )-valued random vector Y = {y i } n i=1; y i = n u ij x j C(O n ) L (Ω, P ) j=1

13 1.1. MOTIVATION 5 satisfies the invariance condition (id τ)q(y 1,..., y n ) = τ ( Q(x 1,..., x n ) ) 1 C(On), (1.1.1) for any polynomial Q in n (non-commuting) variables. Moreover, there is a converse: any distribution-preserving linear transformation X U X of a real Gaussian n-vector must correspond to an orthogonal matrix U O n. In summary, the distributional symmetries of a standard Gaussian n-vector are encoded by the orthogonal group O n. In the context of free probability, the natural analogue of the Gaussian vector considered above is given by an n-tuple X = {x i } n i=1 of identically distributed and freely independent semicircular operators in a tracial von Neumann algebra (M, τ) (see [72, Theorem 8.17]). In this setting, it is natural to ask what types of symmetries this non-commutative random vector can have? To answer this question, we mimic the situation for Gaussian vectors and look for transformations X Y = {y i } n i=1 of the form y i = n u ij x j A M, j=1 where A is a (possibly non-commutative) unital -algebra generated by coordinate elements {u ij } 1 i,j n. For such a transformation to define a symmetry of the semicircular family X, the natural requirement is that equation (1.1.1) hold with the algebra C(O n ) replaced by the algebra A. By carefully analyzing equation (1.1.1) (see [34] for details), one can conclude that the only relations imposed on the generators {u ij } 1 i,j n A are that they form an orthogonal matrix U = [u ij ] M n (A).

14 1.1. MOTIVATION 6 That is, the generators {u ij } 1 i,j n satisfy the same relations as the coordinate functions on O n, except that they do not necessarily commute! This observation shows that the free analogue of a Gaussian vector admits more symmetries than its classical counterpart. In particular, since the coordinate functions {u ij } A describing these symmetries do not commute, we can t encode them with a classical symmetry group like O n. Instead, we need a more general object a quantum group. The preceding analysis naturally leads us to the free orthogonal quantum group O + n. O + n is an abstract group-like object whose non-commutative algebra of continuous functions, C u (O + n ), is given by the universal C -algebra generated by operators {u ij } 1 i,j n subject to the relations which make U = [u ij ] M n (C u (O + n )) an orthogonal matrix. The object O + n turns out to satisfy Woronowicz axioms [103] for a compact quantum group. In particular, the quantum group O + n admits a Haar state (a non-commutative analogue of the Haar measure). Using this Haar state, we can build a left regular representation λ in much the same way that was outlined earlier for a discrete group Γ using the trace τ : C[Γ] C. From the representation λ, we obtain the reduced C -algebra C r (O n + ) = λ(c u (O n + )) and reduced von Neumann algebra L (O n + ) = λ(c u (O n + )). The operator algebras C r (O n + ) and L (O n + ) turn out to be very interesting and concrete operator algebras, which behave in many respects like the algebras Cr (F n ) and V N(F n ) associated to the free group F n. For other classes of compact matrix Lie groups G including the permutation group S n, the unitary group U n, and automorphism groups of finite dimensional C -algebras, a similar liberation procedure can be applied to the coordinate functions in C(G), yielding further examples of free quantum groups G. In many cases, these free quantum groups can also be realized as quantum symmetries of freely independent random

15 1.2. THESIS ORGANIZATION AND OVERVIEW OF MAIN RESULTS 7 variables (see [15, 34, 61]). This thesis studies these free quantum groups and the reduced operator algebras C r (G) and L (G) associated to them. Our principal aim is to uncover and highlight various structural similarities between these quantum group operator algebras and those operator algebras associated to the free groups. It is hoped that the results obtained in this work will open up new directions of research on the structure of free quantum group operator algebras, and perhaps even provide further insight into free probability theory and the structure of the free group factors. 1.2 Thesis Organization and Overview of Main Results Aside from the present chapter, this thesis consists of five additional chapters. Chapter 2 is expository in nature and contains most of the preliminaries on operator algebras, quantum groups, and free probability that will be required in this thesis. Chapters 3, 4 and 5 present the original results of this thesis. These three chapters are based on the articles [25], [24] and [26], respectively. Finally, Chapter 6 is a very short chapter containing some concluding remarks and a list of open problems for future research. Let us now briefly outline the main results of Chapters 3-5. Any unexplained terminology will be properly defined in the following chapters. Chapter 3: In this chapter, we formally introduce the free orthogonal and free unitary quantum groups G = O n +, U n +, respectively, and investigate their operator algebraic structure. From results in [3, 90, 91, 93], it is known that the von Neumann algebras L (G) are generally solid and prime II 1 -factors, and that the reduced C - algebras C r (G) are exact and simple with unique trace. All of these remarkable structural properties are shared with the free group operator algebras V N(F k ) and

16 1.2. THESIS ORGANIZATION AND OVERVIEW OF MAIN RESULTS 8 Cr (F k ). Moreover, in [3] it is shown (using free probability techniques) that there exists an isomorphism L (U 2 + ) = V N(F 2 ). Based on the above analogies with free group operator algebras, it is natural to ask what other properties of the free group operator algebras are shared with L (G) and C r (G)? In Chapter 3, we study approximation properties for the operator algebras L (G) and C r (G). The main result of the chapter is the establishment of the Haagerup approximation property for L (G), answering questions posed by Vaes and Vergnioux in [89], [93] and [94]. For the free group factors V N(F k ), the Haagerup approximation property was proved by Haagerup in his groundbreaking work [52]. In the free group context, this property amounts to the fact that the family of functions {ϕ t } t>0 c 0 (F k ); ϕ t (g) = e g t (g F k ), forms an approximate identity for the algebra c 0 (F k ) (of complex functions vanishing at infinity), consisting of positive definite functions. Here, : F k [0, ) denotes the canonical length function associated to a fixed set of free generators {g 1,..., g k } F k. By combining the Haagerup approximation property with certain norm-inequalities provided by the property of rapid decay for the dual quantum groups Ĝ (proved by Vergnioux in [93]), we also establish the metric approximation property for the C - algebras C r (G). Using the metric approximation property together with a duality argument, we obtain an interesting structure result for the quantum convolution algebra L 1 (G). Namely, we prove that, although L 1 (G) turns out to never have a bounded approximate identity, it always admits an unbounded approximate identity which is uniformly bounded in the multiplier norm on L 1 (G). This result is again

17 1.2. THESIS ORGANIZATION AND OVERVIEW OF MAIN RESULTS 9 reminiscent of the situation one has for free groups. Indeed, regarding L 1 (G) as the quantum analogue of the Fourier algebra A(Γ) of a discrete group Γ (see [47] for the definition of A(Γ)), our result parallels Haagerup s remarkable discovery ([52]) of the existence of multiplier-bounded approximate identities for the Fourier algebras A(F k ). It is interesting to note that the free orthogonal and free unitary quantum groups (along with some other examples studied in Chapter 4) currently appear to be the only genuinely quantum examples of compact quantum groups where L 1 (G) fails to have a bounded approximate identity, yet admits a multiplier-bounded one. (By genuinely quantum, we mean here that G is not the dual of a discrete group, and does not arise as a free product of co-amenable compact quantum groups.) Chapter 4: Let B be a finite dimensional C -algebra and let ψ : B C be the canonical trace on B given by restricting to B the unique trace on L(B) (where B L(B) via its left-regular representation). In this chapter, we study the tracepreserving quantum automorphism group G = G aut (B, ψ). Within the framework of non-commutative geometry, these quantum groups simultaneously play the role of the classical permutation groups of finite sets and the compact Lie groups of - automorphisms of finite-dimensional C -algebras. The principal aim of Chapter 4 is to study the operator algebras L (G) and C r (G), associated to the quantum groups G = G aut (B, ψ). It is generally believed that these operator algebras should share many structural properties with the free unitary and free orthogonal quantum group operator algebras studied in Chapter 3 (and therefore also the free group operator algebras) see [7, 12]. Using the techniques that we develop in Chapter 3, we prove that the von Neumann algebras L (G) always have the Haagerup approximation property. Next, we consider the discrete

18 1.2. THESIS ORGANIZATION AND OVERVIEW OF MAIN RESULTS 10 dual quantum groups Ĝ and prove that they have the property of rapid decay, using techniques inspired by Vergnioux [93]. The property of rapid decay is then used in conjunction with the Haagerup approximation property to prove that the reduced C -algebras C r (G) always have the metric approximation property, and that the convolution algebras L 1 (G) have multiplier-bounded approximate identities. We then go on to study some algebraic properties of L (G) and C r (G). Using the representation theory of G and planar algebra techniques, we show that if dim B 8, then L (G) is a full type II 1 -factor and C r (G) is exact and simple with unique trace. These factoriality and simplicity results partially answer conjectures of Banica and Collins in [12] and [7]. We also show that if B = M n (C) is a full matrix algebra (with n 3), then L (G) embeds into L (O n + ) as an index 2 subfactor. This allows us to deduce (in this special case) that L (G) solid and prime. Chapter 5: In this second last chapter, we switch gears slightly and consider certain functional analytic inequalities which occur in the context of free probability theory. More precisely, we study the strong Haagerup inequality for -free R-diagonal operators, due to Kemp and Speicher [60]. Given a -free identically distributed family of R-diagonal operators {x r } r Λ in a tracial C -probability space (A, ϕ), the strong Haagerup inequality of Kemp-Speicher is an L 2 L norm equivalence of the form T L 2 (A,ϕ) T A C d T L 2 (A,ϕ), for any homogeneous polynomial T = a i x i(1) x i(2)... x i(d) A i:{1,...,d} Λ

19 1.2. THESIS ORGANIZATION AND OVERVIEW OF MAIN RESULTS 11 of degree d in the variables {x r } r Λ. Here, C > 0 is a universal constant depending only on the common distribution of the variables {x r } r Λ. The goal of this chapter is to generalize this strong Haagerup inequality to a broader context by replacing the condition of free independence on the family {x r } r Λ with the (strictly weaker) property of having a joint -distribution which remains invariant when acted on by certain compact quantum groups. More precisely, we consider families of operators {x r } r Λ (A, ϕ), whose joint -distribution is invariant under both the action of the torus T by free complexification and the natural action of the hyperoctahedral quantum groups {H n + } n 1. We prove a strong form of Haagerup s inequality for the non-self-adjoint operator algebra B generated by {x r } r Λ, which generalizes the Kemp-Speicher strong Haagerup inequality. As an application of our result, we show that B always has the metric approximation property. We also apply our techniques to study the reduced C -algebra of the free unitary quantum group U n +. We show that the non-self-adjoint subalgebra B n generated by the matrix elements of the fundamental representation of U + n satisfies a strong form of Haagerup s inequality, improving on the estimates given by Vergnioux s property of rapid decay for the dual quantum groups Û + n [93].

20 12 Chapter 2 Background and Notation In this chapter we briefly overview some of the notation and concepts from operator algebras, free probability and quantum group theory that will be used in this thesis. Throughout this thesis, we will use the convention that the natural numbers N include 0, and that [m] := {1, 2,..., m} for any m N\{0}. For the convenience of the reader, we point out that the material in Section 2.3 on free probability will (for the most part) only be required for the results of Chapter Operator Algebras Our main references for this section are the books [76], [77] and [85]. All vector spaces considered in this thesis will be over the field C. Given a Banach space X, L(X) will denote the algebra of all linear operators T : X X and B(X) L(X) will denote the Banach algebra of bounded linear operators (equipped with the operator norm T = sup x =1 T x ). The identity of B(X) will be written as id X. The dual space of norm-continuous linear functionals on X will be denoted by X. If x X and ϕ X, we will typically write x, ϕ for the evaluation of ϕ at x.

21 2.1. OPERATOR ALGEBRAS 13 For a Hilbert space H, we denote the inner product of two vectors η, ξ H by η ξ, and always take this inner product to be conjugate-linear in the first variable. Let T T be the usual antilinear isometric involution on B(H) defined by η T ξ = T η ξ, (T B(H), ξ, η H). A (concrete) C -algebra is a norm-closed -subalgebra A B(H). The weak operator topology (WOT) on B(H) is the topology on B(H) defined by the family of semi-norms T η T ξ, (η, ξ H). Given η, ξ H, we write ω η,ξ B(H) for the WOT-continuous linear functional T η T ξ. A von Neumann algebra is a WOT-closed unital -subalgebra A B(H). By von Neumann s famous bicommutant theorem, a -subalgebra A B(H) is a von Neumann algebra if and only if A = A := (A ), where P = {S B(H) : ST = T S T P } for any P B(H). Note that every von Neumann algebra is automatically a C -algebra, since the WOT is weaker than the norm-topology on B(H). We remark that it is possible (and often extremely useful) to define C -algebras and von Neumann algebras intrinsically as certain classes of abstract involutive Banach algebras. The Gelfand-Naimark-Sakai-Segal constructions then show that every abstract C -algebra or von Neumann algebra can be isometrically represented as a concrete C -algebra or von Neumann algebra acting on some Hilbert space H, as defined above Tensor Products Let A and B be -algebras, and denote by A alg B their algebraic tensor product. A alg B is a -algebra in the obvious way. If A B(H) and B B(K) are C - algebras acting on Hilbert spaces H and K, respectively, then their minimal tensor

22 2.1. OPERATOR ALGEBRAS 14 product is the C -algebra A B := A alg B B(H K) B(H K), where H K is the Hilbert space tensor product of H and K. If, moreover, A and B are von Neumann algebras, their spatial tensor product is the von Neumann algebra A B := A alg B WOT = (A alg B) B(H K). For operators on multiple tensor products of Hilbert spaces, we recall the standard leg numbering notation. For example, if H 1, H 2, H 3 are Hilbert spaces and T B(H 1 H 2 ), then T 12 := T id H3 B(H 1 H 2 H 3 ) is the operator which acts by T on the first two factors of the tensor product and as the identity map on the third factor. Similarly, T 23 = id H3 T B(H 3 H 1 H 2 ). Finally, T 13 B(H 1 H 3 H 2 ) is the operator which acts by T on the first and third factors of H 1 H 3 H 2 and by the identity map on the second factor. Explicitly, T 13 = (id σ )T 12 (id σ) B(H 1 H 3 H 2 ) where σ : H 3 H 2 H 2 H 3 is the unitary tensor flip map Completely Bounded Maps Let A B(H) be a C -algebra. For each integer n 1, let l 2 (n) be an n-dimensional Hilbert space and identify M n (C) with B(l 2 (n)). Then the -algebra M n (A) = M n (C) alg A of n n matrices over A can be given a unique C -algebra structure by identifying M n (A) = M n (C) A B(l 2 (n) H).

23 2.1. OPERATOR ALGEBRAS 15 Let n : M n (A) [0, ) be the norm on M n (A) coming from the above identification. Given a linear map Φ : A A, the nth amplification of Φ is the linear map Φ (n) : M n (A) M n (A) defined by Φ (n) = id l 2 (n) Φ under the above identification. We say that Φ is completely bounded (CB) if Φ cb := sup Φ (n) Mn(A) Mn(A) <, n 1 Φ is a a complete contraction if Φ cb 1, and Φ is completely positive (CP) if each amplification Φ (n) : M n (A) M n (A) is positive. (Recall that if B is a C - algebra, then Φ : B B is positive if and only Φ maps the cone of positive operators into itself.) Note that if A is unital and Φ : A A is CP, then Φ is CB with Φ cb = Φ(1 A ) (see [76], Proposition 3.6). The algebra of completely bounded maps on A is denoted by CB(A), and becomes a Banach algebra with respect to the norm cb. Let A B(H) be a von Neumann algebra. A fundamental theorem of Sakai (see [85], Chapter 3) states that there exists a Banach space A (called the predual of A), such that A is isometrically isomorphic to the dual Banach space (A ). Moreover, A is uniquely determined up to isometric isomorphism. The weak- topology on A induced by this duality is called the σ-weak topology on A. Elements of A, when viewed as linear functionals on A under the canonical embedding A A, are called normal linear functionals. A linear map Φ B(A ) is called completely bounded if its linear adjoint Φ belongs to CB(A). The collection of such maps is denoted by CB(A ). A map Φ B(A) is called σ-weakly continuous (or normal) if there exists a (necessarily unique) Φ B(A ) such that Φ = (Φ ). One can also consider normal,

24 2.1. OPERATOR ALGEBRAS 16 completely bounded maps between different von Neumann algebras. We leave the (obvious) modifications of the above definitions to the reader. Example Let A and B be C algebras. A -homomorphism Φ : A B is always completely positive. Moreover, if ϕ A is a bounded linear functional, then the left slice map Φ ϕ : A B B; Φ ϕ = ϕ id B, is completely bounded with Φ ϕ cb = ϕ The GNS Representation and L 2 -Extensions of CP Maps Let A be C -algebra with unit 1 A. A state on A is a linear functional ϕ : A C, such that ϕ(1 A ) = 1 and ϕ(a a) 0 (a A). We call ϕ faithful if for each a A, ϕ(a a) = 0 a = 0. Note that a state is automatically continuous, with ϕ = ϕ(1 A ) = 1. Each state ϕ : A C determines a sesquilinear form ϕ on A by setting a b ϕ = ϕ(a b), (a, b A). Let N ϕ = {a A : ϕ(a a) = 0}. Then N ϕ is a closed left ideal of A, and ϕ factors through the quotient to define a non-degenerate sesquilinear form on A/N ϕ, which we still denote by ϕ. Let Λ ϕ : A A/N ϕ be the canonical map and denote the Hilbert space completion of Λ ϕ (A) = A/N ϕ by L 2 (A, ϕ) (or just L 2 (A) if ϕ is understood). The Gelfand-Naimark-Segal (GNS) representation is the -homomorphism π ϕ : A B(L 2 (A, ϕ)) which canonically extends the left action of A on itself by multiplication: π ϕ (a)λ ϕ (b) = Λ ϕ (ab) (a, b A).

25 2.1. OPERATOR ALGEBRAS 17 A simple calculation shows that ϕ(a) = ξ 0 π ϕ (a)ξ 0 ϕ for each a A, where ξ 0 = Λ ϕ (1 A ). That is, the state ϕ can be recovered from the GNS representation π ϕ through the normal vector state ω ξ0,ξ 0 B(L 2 (A, ϕ)). Note that if we assume that ϕ is a faithful state, then the GNS representation π ϕ : A B(L 2 (A, ϕ)) is faithful (i.e., injective). In this case, we can, without loss of generality, identify A with the concrete C -subalgebra π ϕ (A) B(L 2 (A, ϕ)). Moreover, Λ ϕ : A L 2 (A, ϕ) is an injective map with dense range in this case, allowing us to identify A with the dense subspace Λ ϕ (A) L 2 (A, ϕ). In this situation, we often abuse notation and write x 2 or x L 2 (A) for the quantity Λ ϕ (x) L 2 (A,ϕ). Let A and B be unital C -algebras, equipped with faithful states ϕ : A C and ψ : B C, respectively, and let Φ : A B be a CP map. It follows from the preceding discussion that we can view Φ as a (possibly only densely defined) linear map ˆΦ : Λ ϕ (A) L 2 (A, ϕ) L 2 (B, ψ); ˆΦ(Λϕ (a)) = Λ ψ (Φ(a)) (a A). The following theorem shows that, under certain reasonable assumptions on Φ, ˆΦ can be extended to a bounded linear map L 2 (A, ϕ) L 2 (B, ψ). The assumption that we need (for our purposes) is that Φ : A B is a ϕ ψ-markov map. That is, Φ preserves the given states: ψ Φ = ϕ. Theorem Let A, B, ϕ, ψ and Φ be as described above. If Φ is a ϕ ψ-markov map, then ˆΦ extends to a bounded linear map L 2 (A, ϕ) L 2 (B, ψ) with ˆΦ 1 Proof. The proof of this result relies on the Schwartz inequality for CP maps (see

26 2.1. OPERATOR ALGEBRAS 18 [76], Proposition 3.3), which says that Φ(a) Φ(a) Φ(a a) (a A). Using this inequality, we have (for each a A), ˆΦ(Λ ϕ (a)) 2 L 2 (B,ψ) = Λ ψ(φ(a)) 2 L 2 (B,ψ) = ψ(φ(a) Φ(a)) ψ(φ(a a)) = ϕ(a a) = Λ ϕ (a) 2 L 2 (A,ϕ), showing that ˆΦ 1 and the result follows. We call the map ˆΦ : L 2 (A, ϕ) L 2 (B, ψ) the L 2 -extension of Φ Nuclearity, Injectivity, and II 1 -Factors For C -algebras and von Neumann algebras, the notions of nuclearity and injectivity are extremely important and have many equivalent formulations. For our purposes, we take the following formulations: A C -algebra A is called nuclear if there exists a net {Φ λ } λ Λ of finite rank, contractive completely positive maps Φ λ : A A, such that lim Φ λ (a) a = 0 (a A). λ A von Neumann algebra A is called injective if there exists a net {Φ λ } λ Λ of normal, unital, finite rank completely positive maps Φ λ : A A, such that lim λ Φ λ (a) = a (σ-weakly a A). Let A be a von Neumann algebra (with seperable predual A ). We say that A

27 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 19 is finite if there exists a normal, tracial state τ A. (Recall that tracial means τ(ab) = τ(ba) for all a, b A). A von Neumann algebra A is called a factor if its centre is trivial: Z(A) = C1 A. If a factor A is also finite, we say that A is a II 1 -factor. It is a standard fact that a finite von Neumann algebra A is a II 1 -factor if and only if there exists a unique tracial state τ A. 2.2 Compact and Discrete Quantum Groups We present here an overview of the basic facts on compact and discrete quantum groups that will be needed for this thesis. Our main reference for this will be [103] and the excellent book [86]. Definition A compact quantum group is a pair G = (A, ) where A is a unital C -algebra, : A A A is a unital -homomorphism satisfying the co-associativity relation (id A ) = ( id A ), (2.2.1) and G satisfies the cancellation property. That is, the sets (A)(1 A A), and (A)(A 1 A ), span dense subspaces of A A. From these three axioms, it follows that every compact quantum group G = (A, ) admits a Haar state, which is a state h : A C defined uniquely by the following

28 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 20 bi-invariance condition with respect to the coproduct: (h id A ) (a) = (id A h) (a) = h(a)1 A (a A). (2.2.2) Let L 2 (G) := L 2 (A, h) be the Hilbert space obtained from the GNS construction applied to the sesquilinear form on A defined by a b h = h(a b), and let λ : A B(L 2 (G)) be the associated GNS representation (which we call the left regular representation). Recall that λ is given by λ(a)λ h (b) = Λ h (ab), where a, b A and Λ h : A L 2 (G) is the GNS map. The central objects of study in this thesis are the reduced C -algebra and reduced von Neumann algebra of G. These are the concrete operator algebras acting on the Hilbert space L 2 (G) given by C r (G) := λ(a) B(L 2 (G)) and L (G) := C r (G) B(L 2 (G)), respectively. Since h = (h h) (i.e., is an h h h-markov map), it follows that descends to a coproduct r : C r (G) C r (G) C r (G) given by r λ = (λ λ). Furthermore, r extends to a normal -homomorphism r : L (G) L (G) L (G). The pair (C r (G), r ) then becomes a compact quantum group (called the reduced version of G) and (L (G), r ) becomes a von Neumann algebraic compact quantum group, in the sense of [66]. The (normal, faithful) Haar state h r : L (G) C is determined by h r (λ(a)) = h(a) for each a A. Faithfulness of h r on L (G) follows from the fact that h is a KMS state see [86, Example ]. From now on we will drop the subscript r from h r, and just write h for the Haar state in all possible situations. No

29 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 21 confusion should arise from this. The pre-adjoint of r induces a completely contractive Banach algebra structure on the predual L 1 (G) := L (G). L 1 (G) is called the convolution algebra of G. G is said to be of Kac-type if the Haar state h is tracial. In this case, L (G) is a finite von Neumann algebra with finite Haar trace h. For any compact quantum group G = (A, ), the Banach space A of continuous linear functionals on A is endowed with an associative convolution product (ϕ, ψ) ϕ ψ = (ϕ ψ). With this algebra structure, A becomes a completely contractive Banach algebra and the convolution algebra L 1 (G) can be completely isometrically identified with a closed two sided ideal in A via the dual pairing a, ω = ω, λ(a) (a A, ω L 1 (G)). (2.2.3) See [65, Page ] for details Basic Examples As the terminology introduced above suggests, compact quantum groups should be thought of as natural non-commutative geometric generalizations of classical compact groups. This first example shows that they do indeed generalize the notion of a compact group. Example Let G be a compact group, let A = C(G) be the C -algebra of continuous functions on G (with -algebra structure given by the pointwise product and complex conjugation of functions), and let : C(G) C(G G) = C(G) C(G)

30 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 22 be the unital -homomorphism which dualizes the group product: f(x, y) = f(xy) (f C(G), x, y G). Then the pair G = (A, ) is a compact quantum group. Indeed, the co-associativity of just reduces to the fact that the group law is associative, and the cancellation property for turns out to be equivalent to the fact that multiplication in G has both the left and right cancellation property (see [86], Proposition 5.1.3). Let dg denote the unique Haar probability measure on G. Then the Haar state h : A C is given by integration with respect to dg: h(f) = f(g)dg G (f A). Doing the GNS construction with respect to h yields all of the familiar objects: L 2 (G) = L 2 (G, dg), C r (G) = C(G) and L (G) = L (G, dg) (both viewed as multiplication operators acting on L 2 (G, dg)), and L 1 (G) = L 1 (G, dg). By the Reisz representation theorem, the Banach space dual A = C(G) can be identified with M(G), the Banach space of complex regular Borel measures on G. The convolution product on M(G) is just the usual convolution of measures on G. Remark Note that in the above example, the Haar state is always faithful, and therefore C r (G) = A. As the following example shows, this faithfulness of the Haar state (and GNS representation) is not always guaranteed in the non-commutative setting. Example Let Γ be a discrete group and let C (Γ) denote the universal enveloping C -algebra of Γ. C (Γ) is completion of the group algebra C[Γ] with respect to

31 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 23 the C -algebra norm f C (Γ) = sup σ σ(f), where the supremum runs over all - representations σ : C[Γ] B(H). For more details, see [85], Vol. 1, Page 45. There is a natural coproduct : C (Γ) C (Γ) C (Γ) given by (π(γ)) = π(γ) π(γ) (γ Γ), where π : C[Γ] C (Γ) is the natural embedding. The pair G = (C (Γ), ) then satisfies the axioms of a compact quantum group. The Haar state on C (Γ) is just the (extension to C (Γ) of the) canonical trace τ : C[Γ] C; τ(γ) = δ γ,e, (γ Γ). The GNS construction then yields the natural identifications L 2 (G) = l 2 (Γ), C r (G) = Cr (Γ), L (G) = V N(Γ), and L 1 (G) = A(Γ), the Fourier algebra of Γ. (Recall that A(Γ) = V N(Γ) can be identified with the space of coefficient functions of the left regular representation λ : Γ U(l 2 (Γ)), and the algebra structure on A(Γ) is just the pointwise product of functions.) Note that if Γ is abelian with compact Pontryagin dual group G = ˆΓ, then the Fourier transform yields an identification between the compact quantum group (C (Γ), ) and the one constructed from G = ˆΓ as in Example In the case of a general discrete group Γ, we note the important fact that the quotient map C (Γ) Cr (Γ) given by the GNS representation is faithful if and only if the Haar trace τ : C (Γ) C is faithful if and only if Γ is amenable. Finally, we briefly mention an important non-classical example. This is Woronowicz so-called q-deformation of SU(2) (see [101]). Example Let q [ 1, 1] and let A be the universal C -algebra generated by

32 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 24 elements α, γ subject to the relations α α + γ γ = 1, αα + q 2 γγ = 1, γ γ = γγ, qγα = αγ, qγ α = αγ. If we define a unitary U := u 11 u 12 u 21 u 22 = α γ qγ α M 2 (A), then there is a coproduct : A A A defined uniquely by u ij = 2 k=1 u ik u kj, (1 i, j 2). The pair SU q (2) = (A, ) then becomes a compact quantum group. Note that when q = 1, we recover the classical group SU(2) as in Example In this example, it can be shown that the Haar state h : A C is always faithful, but tracial only when q = ± Representation Theory Let H be a finite dimensional Hilbert space. A representation of a compact quantum group G = (A, ) on H is an element U B(H) A such that (id )U = U 12 U 13. (2.2.4) By fixing an orthonormal basis {e i } d i=1 for H and writing U = [u ij ] M d (A) relative to this basis, (2.2.4) is equivalent to requiring that (u ij ) = d k=1 u ik u kj for 1 i, j d. U is called a unitary representation if U is, in addition, a unitary element of M d (A). If U 1 = [u 1 i(1)j(1) ] M d(1)(a) and U 2 = [u 2 i(2)j(2) ] M d(2)(a) are (unitary) representations, then their tensor product U 1 U 2 = [u 1 i(1)j(1) u2 i(2)j(2) ] M d(1)d(2)(a)

33 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 25 and direct sum U 1 U 2 M d(1)+d(2) (A) are also (unitary) representations. The vector space Mor(U 1, U 2 ) = {T B(H 1, H 2 ) (T 1 A )U 1 = U 2 (T 1 A )}, is called the space of morphisms (or intertwiners) between U 1 and U 2. U 1 and U 2 are called (unitarily) equivalent if there exists an invertible (unitary) operator T Mor(U 1, U 2 ), and in this case we write U 1 = U 2. If U = [u ij ] M d (A) is a representation, then so is U = [u ij]. The representation U is called the conjugate of U. Note that the passage from U to its conjugate U does not preserve unitarity in general. A representation U is called irreducible if Mor(U, U) = Cid. Note that U is irreducible if and only if U is irreducible. The following is a fundamental result in the theory of compact quantum groups. Theorem Every irreducible representation of a compact quantum group G = (A, ) is finite dimensional and equivalent to a unitary one if it is invertible. Moreover, every unitary representation is unitarily equivalent to a direct sum of irreducibles. Let {U α = [u α ij] B(H α ) A : α Irr(G)} be a maximal family of pairwise inequivalent finite-dimensional irreducible unitary representations of G = (A, ), with 0 Irr(G) denoting the index corresponding to the trivial representation 1 A A = M 1 (A) and U α denoting the representative of the equivalence class of U α. If A A denotes the linear span of {u α ij : 1 i, j d α, α Irr(G)}, then A is a Hopf -algebra on which h is faithful, A is norm dense in A, and the set {u α ij : 1 i, j d α, α Irr(G)} is a linear basis for A. (We refer the reader to Chapter 3 of [86] for the basics on Hopf -algebras.) The coproduct : A A alg A is just the

34 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 26 restriction A, the coinverse κ : A A is the antihomomorphism given by κ(u α ij) = (u α ji) (α Irr(G), 1 i, j d α ), and the counit ɛ : A C is the -character given by ɛ(u α ij) = δ ij (α Irr(G), 1 i, j d α ). The Hopf -algebra A is unique in the sense that if B A is any other dense Hopf -subalgebra, then B = A (see [17, Theorem 5.1]). In general, the counit ɛ : A C and the coinverse κ : A A defined above cannot be extended to bounded linear maps on A. A compact quantum group G = (A, ) is called co-amenable if the map λ(a) ɛ(a) (a A) extends to a character of C r (G). (Note that a compact quantum group G = (C (Γ), ) associated to a discrete group Γ is co-amenable iff Γ is amenable.) In this thesis we will mainly deal with compact quantum groups of Kac type, which turn out to be precisely those for which the coinverse κ : A A can be extended to a bounded map on A. Definition A compact quantum group G = (A, ) is of Kac type if any one of the following equivalent conditions is satisfied. 1. The Haar state h : A C is a trace. 2. κ : A A has a continuous extension to a -antihomomorphism κ : A A. 3. κ 2 = id A. See [1] for proofs of the above equivalences.

35 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 27 When G = (A, ) is of Kac type and U α = [u α ij] B(H α ) A (α Irr(G)) is an irreducible unitary representation, it is readily checked that U α is also unitary. We therefore assume in the Kac case that the representatives {U α : α Irr(G)} have been chosen so that U α = U α. In this case, we have κ(u α ij) = (u α ji) = u α ji (1 i, j d α, α A). (2.2.5) Finally, when G = (A, ) is of Kac type, the map κ r : C(G) C(G) defined by κ r (λ(a)) = λ(κ(a)) (a A), defines the coinverse for the reduced quantum group (C r (G), r ). Furthermore, κ r extends to a normal -antiautomorphism κ r : L (G) L (G) Orthogonality Relations Let G = (A, ) be a compact quantum group and {U α = [u α ij] B(H α ) A : α Irr(G)} a fixed family of irreducible unitary representation representatives as above. For each α Irr(G), there exists an isometric morphism t α Mor(1, U α U α ) (which, by irreducibility, is unique up to scalar multiplication by T). Let j α : H α H α be the conjugate-linear invertible map defined by j α ξ = (ξ id)t α (ξ H α ), (invertibility of j α follows from the irreducibility of U α ), and let Q α = j αj α > 0. By renormalizing j α and j α, we may assume that Tr(Q α ) = Tr(Q 1 α ) and that j α j α = id Hα Mor(U α, U α ). Using the operators {Q α } α Irr(G), we can describe the Schur orthogonality relations for the matrix elements of irreducible unitary representations

36 2.2. COMPACT AND DISCRETE QUANTUM GROUPS 28 of G: h((u α ij) u β kl ) = δ αβδ jl (Q 1 α ) ki TrQ α and h(u α ij(u β kl ) ) = δ αβδ ik (Q α ) lj TrQ α, where α, β Irr(G) and 1 i, j d α, 1 k, l d β. See [103, Section 6] for a proof of this result. In particular, setting L 2 α(g) = span{λ h (u α ij) : 1 i, j d α }, we obtain the following orthogonal decomposition of L 2 (G). L 2 (G) = L 2 α(g) = H α H α, (2.2.6) α Irr(G) α Irr(G) where the second isomorphism is given on each direct summand L 2 α(g) by Λ h ( (ωη,ξ id A )U α) ξ (1 η )t α (α Irr(G), ξ, η H α ). (2.2.7) For future reference, we will always write P α for the orthogonal projection from L 2 (G) onto L 2 α(g) = H α H α. Note that when G is of Kac-type, traciality of the Haar state combined with the preceding orthogonality relations forces Q α = id Hα {d 1/2 α Λ h (u α ij) : 1 i, j d α } is an orthonormal basis for L 2 α(g). for each α Irr(G). Thus Universal Quantum Groups Let G = (A, ) be a compact quantum group and let G r = (C r (G), r ) be the reduced version of G. Note that G r is in a sense, the minimal realization of G, since C r (G) is always a quotient of A. On the other end of the spectrum, there also exists a maximal realization of G: Let C u (G) be the universal enveloping C -algebra of the Hopf - algebra A and let π u : A C u (G) be the universal representation. Then there exists