Discrete quantum groups and their probabilistic boundaries Stefaan - PDF document

Fields Institute Summer School Operator Algebras – August 2007 – Ottawa Discrete quantum groups and their probabilistic boundaries Stefaan Vaes, K.U.Leuven, stefaan.vaes@wis.kuleuven.be Disclaimer. Due to lack of time, these lecture notes are in very premature form and probably con- tain a lot of errors and misprints. I would be glad to hear any of your questions/comments/suggestions. Exercises and discussions. I included several exercises in the text. They are not meant to be very deep, but rather to train you and to let you understand how things work. If some people are interested in asking questions or discussing the exercises, this can be organized in the afternoon. Please come and see me. Some references. The following is by no means an exhaustive list of references , but rather a guide to enter the recent literature on compact or discrete quantum groups from the point of view taken in the course. References [1] A very good beginners’ introduction to the theory of compact and discrete quantum groups. A. Maes & A. Van Daele , Notes on compact quantum groups. Nieuw Arch. Wisk. (4) 16 (1998), 73–112. arXiv:math/9803122 [2] The paper where it all started from, published in 1998, but written in the beginning of the 1990’s. S.L. Woronowicz , Compact quantum groups. In Sym´ etries quantiques (Les Houches, 1995) , North- Holland, Amsterdam, 1998, pp. 845–884. Available on http://www.fuw.edu.pl/ ∼ slworono/Prace.html [3] Computation of the representation theory of the compact quantum group A o ( F ). T. Banica , Th´ eorie des repr´ esentations du groupe quantique compact libre O ( n ). C. R. Acad. Sci. Paris S´ er. I Math. 322 (1996), 241–244. Available on http://picard.ups-tlse.fr/ ∼ banica/pub.html [4] Computation of the representation theory of the compact quantum group A u ( F ). T. Banica , Le groupe quantique compact libre U( n ). Commun. Math. Phys. 190 (1997), 143–172. Available on http://picard.ups-tlse.fr/ ∼ banica/pub.html [5] Construction of ergodic actions of the quantum groups SU q (2), A o ( F ) and A u ( F ) with remarkable properties. The main tool is the construction of a link C ∗ -algebra between monoidally equivalent quan- tum groups. J. Bichon, A. De Rijdt & S. Vaes , Ergodic coactions with large multiplicity and monoidal equiv- alence of quantum groups. Comm. Math. Phys. 262 (2006), 703–728. arXiv:math/0502018 [6] The tool of [5] is used to describe how the Poisson boundary behaves when passing to a monoidally equivalent quantum group. In combination with [9], this allows to compute the Poisson boundary of a large class of quantum groups, including the duals of A o ( F ). A. De Rijdt & N. Vander Vennet , Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries. Preprint. arXiv:math/0611175 [7] The first appearence of random walks and their Poisson boundaries in quantum group theory, moti- vated by the search for minimal actions and their relations to subfactors. The actual computation in the paper of the Poisson boundary of the dual of SU q (2) has since then be simplified in [8] and even 1

more in [9] recently. M. Izumi , Non-commutative Poisson boundaries and compact quantum group actions. Adv. Math. 169 (2002), 1–57. [8] The paper computes in a very elegant way what the title says. Since then, a simpler approach became available in [9]. M. Izumi, S. Neshveyev & L. Tuset , Poisson boundary of the dual of SU q ( n ). Comm. Math. Phys. 262 (2006), 505-531. arXiv:math/0402074 [9] Computation of the Poisson boundary for all coamenable compact quantum groups having commuta- tive fusion rules. This includes all q -deformations of compact Lie groups and as such, generalizes and simplifies the computations of [7] and [8]. R. Tomatsu , A characterization of right coideals of quotient type and its application to classification of Poisson boundaries. Preprint. arXiv:math/0611327 [10] A systematic study of minimal/outer actions of quantum groups and their relations with subfactors. S. Vaes , Strictly outer actions of groups and quantum groups. J. Reine Angew. Math. 578 (2005), 147–184. arXiv:math/0211272 [11] Study of the compact quantum group A o ( F ) from an operator algebra point of view : we are interested in the von Neumann algebra L ∞ ( G ). In certain cases, we obtain prime factors. The main tool is the construction of a Gromov boundary, in order to prove the Akemann-Ostrand property and to deduce Ozawa’s solidity. S. Vaes & R.Vergnioux , The boundary of universal discrete quantum groups, exactness and facto- riality. Duke Math. J. , to appear. arXiv:math/0509706 2

1 Introduction Topic Discrete quantum groups Duality Compact quantum groups Operator algebras Motivation Discrete/compact quantum groups arise as follows : • generalized symmetry : – actions on operator algebras → minimal actions, ergodic actions, → random walks, Poisson boundaries, – non-commutative geometry, • subfactor theory. Several of these aspects will arise during the course. Important notational convention. The symbol ⊗ denotes the minimal tensor product of C ∗ - algebras as well as the tensor product of Hilbert spaces. The tensor product of von Neumann algebras is denoted by ⊗ , while the algebraic tensor product of vector spaces is denoted by ⊗ alg . 2 Definition of a compact quantum group Definition 2.1 (Woronowicz [2]) . A compact quantum group G is a pair ( C ( G ) , ∆) consisting of • a unital C ∗ -algebra C ( G ), • a unital ∗ -homomorphism ∆ : C ( G ) → C ( G ) ⊗ C ( G ), satisfying • co-associativity : (∆ ⊗ id)∆ = (id ⊗ ∆)∆ , (2.1) • left and right cancelation properties : ∆( C ( G ))(1 ⊗ C ( G )) and ∆( C ( G ))( C ( G ) ⊗ 1) (2.2) are total in C ( G ) ⊗ C ( G ). 3

Where does this definition come from. • Let A be an abelian C ∗ -algebra and ∆ : A → A ⊗ A a unital ∗ -homomorphism satisfying (2.1). → compact semi-group G such that A = C ( G ) and ∆ : C ( G ) → C ( G × G ) : ∆( F )( x, y ) = F ( xy ) for all x, y ∈ G . (Note that C ( G ) ⊗ C ( G ) = C ( G × G ).) • Then, (2.2) is equivalent with the left and right cancelation properties. Proof. Suppose that ∆( C ( G ))(1 ⊗ C ( G )) is total in C ( G × G ). This means that functions of the form ( x, y ) �→ F ( xy ) G ( y ) densely span C ( G × G ). If then x 0 y = x 1 y , it follows that H ( x 0 , y ) = H ( x 1 , y ) for all H ∈ C ( G × G ), implying that x 0 = x 1 . Exercise 1. Prove the converse. • A compact semi-group with left and right cancelation properties is a compact group. Proof. We have to prove the existence of a unit element and of inverses as well as the continuity of the inverse operation. Given g ∈ G , we can switch to the commutative compact semi-group generated by g and assume that G is commutative. The closed ideals of G have the finite intersection property. So, take the smallest non-empty closed ideal I ⊂ G . Let g ∈ I . Then, gI ⊂ I so that by minimality, gI = I . Take e ∈ I satisfying ge = g . Then, for all h ∈ G , gh = geh and by cancelation, h = eh for all h ∈ G . Then, keh = kh and hence, ke = k for all k ∈ G . We have found the unit element. Moreover, e ∈ I and hence, G = eG ⊂ I . So, every closed non-empty ideal in G equals G . In particular, gG = G for all g ∈ G , proving the existence of an inverse for every g ∈ G . Continuity of the inverse operation follows from its closedness. Conclusion : compact quantum groups G with abelian underlying C ∗ -algebra C ( G ) correspond to compact groups. → reason for the notation C ( G ), although in general, there is no underlying space G . Example 2.2. Let Γ be a discrete group. Define G = � Γ as • C ( G ) is either C ∗ (Γ) or C ∗ r (Γ), • ∆( u g ) = u g ⊗ u g for all g ∈ Γ. Later : many naturally appearing examples of compact quantum groups. 4

3 Existence and uniqueness of the Haar measure Recall. • A compact group G admits a unique translation-invariant probability measure, called the Haar measure. • Probability measures on a compact space X correspond to states on the C ∗ -algebra C ( X ). • The state space of a unital C ∗ -algebra is compact in the weak ∗ -topology. Theorem 3.1 (Woronowicz [2], proof taken from [1]) . Let G = ( C ( G ) , ∆) be a compact quantum group. Then, C ( G ) admits a unique state h satisfying (id ⊗ h )∆( a ) = h ( a )1 = ( h ⊗ id)∆( a ) for all a ∈ C ( G ) . We call h the Haar state of G . Preparation. The convolution product of two states ω, ρ ∈ C ( G ) ∗ is given by ω ∗ ρ = ( ω ⊗ ρ )∆ . Exercise 2. Convince yourself that this definition of the convolution product, coincides with the usual convolution product of measures in the case G is an ordinary compact group. We have to prove the existence of a state h satisfying ω ∗ h = h = h ∗ ω for all states ω . Proof of Theorem 3.1. Step 1. If ω is a state on C ( G ) , there exists a state h on C ( G ) satisfying ω ∗ h = h = h ∗ ω . It suffices to take a weak ∗ -accumulation point of the sequence of states given by n � 1 ω ∗ k . n k =1 Step 2. If ω and h are states on C ( G ) satisfying ω ∗ h = h = h ∗ ω , then ρ ∗ h = ρ (1) h = h ∗ ρ whenever 0 ≤ ρ ≤ ω . Choose a ∈ C ( G ) arbitrary. Put b = (id ⊗ h )∆( a ). Compute (id ⊗ ω )∆( b ) = (id ⊗ ( ω ∗ h ))∆( a ) = (id ⊗ h )∆( a ) = b . It follows that � � (∆( b ) − b ⊗ 1) ∗ (∆( b ) − b ⊗ 1) ( h ⊗ ω ) = 0 . But then also � � (∆( b ) − b ⊗ 1) ∗ (∆( b ) − b ⊗ 1) ( h ⊗ ρ ) = 0 . Using the Cauchy-Schwartz inequality, we get � � ( h ⊗ ρ ) ( c ⊗ 1)(∆( b ) − b ⊗ 1) = 0 for all c ∈ C ( G ) . 5