We study the C * -algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C * -algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.From the amenability of the boundary action of the dual of G = A o (F ), we deduce that the reduced C *algebra C(G) red is exact and satisfies the Akemann-Ostrand property. In the setting of finite von Neumann algebras, Ozawa [17] showed that the Akemann-Ostrand property implies solidity of the associated von Neumann algebra. Since in general C(G) ′′ red is of type III, we need a generalization of Ozawa's definition (see Section 2) and we deduce that, for G = A o (F ), the von Neumann algebras C(G) ′′ red are generalized solid. In particular, for F the n by n identity matrix, we get a solid von Neumann algebra.In Section 5, we make the link between our boundary B ∞ for the dual of G = A o (F ) and boundaries arising from quantum random walks on G. We construct a harmonic state ω ∞ on B ∞ and identify (B ∞ , ω ∞ ) ′′ with

The half-liberated orthogonal group O * n appears as intermediate quantum group between the orthogonal group O n , and its free version O + n . We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between O * n and U n , a non abelian discrete group playing the role of weight lattice for O * n , and a number of methods inspired from the theory of Lie algebras. We use these results for showing that the discrete quantum group dual to O * n has polynomial growth.

We find the fusion rules for the quantum analogues of the complex reflection groups H s n = Z s ≀ S n . The irreducible representations can be indexed by the elements of the free monoid N * s , and their tensor products are given by formulae which remind the Clebsch-Gordan rules (which appear at s = 1).2000 Mathematics Subject Classification. 16W30 (46L65, 46L87).

We introduce the quantum Cayley graphs associated to quantum discrete groups and study them in the case of trees. We focus in particular on the notion of quantum ascending orientation and describe the associated space of edges at infinity, which is an outcome of the non-involutivity of the edge-reversing operator and vanishes in the classical case. We end with applications to Property AO and K-theory.Brought to you by | North Carolina State University (NCSU) Libraries Authenticated Download Date | 5/17/15 1:02 AM The paper is organized as follows:1. In the first section, we recall some notation and formulae concerning discrete quantum groups and classical graphs.2. The second section is a technical one about fusion morphisms of free quantum groups, and its results are only used in the proofs of Sections 6 and 7. The reader will probably like to skip over this section at first.3. In the third section, we give the definition of the Cayley graphs of discrete quantum groups and state some basic results about them. 4. We then restrict ourselves to the case of Cayley trees. We introduce and characterize this notion in the fourth section, where we also study the natural ascending orientation of such a tree.5. In the fifth section, we study more precisely the space of geometric edges of a quantum Cayley tree and we find that the projection of ascending edges onto geometric ones is not necessarily injective.6. We show more precisely in the sixth section that the obstruction to this injectivity is the existence of a natural space of (geometric) edges at infinity, which vanishes in the classical case. 7. In the seventh section, we equip this space with a natural representation of the free quantum group under consideration, thus turning it into an interesting geometric object on its own. 8. Finally the last section deals with applications, as explained above.Acknowledgments. Most of the results of Sections 2-5 and, in the special case of the orthogonal free quantum groups, of Sections 6, 7 and 8.2, were included in my PhD thesis [17] at the University Paris 7. I would like to thank my advisor, Prof. G. Skandalis, for having directed me to the beautiful paper [8] of Julg and Valette, and for his precious help in the redaction of my thesis.The generalization and continuation of these results, as well as the redaction of the present article, were completed during a one-year stay at the University of Mü nster, where I could benefit from a postdoctoral position. It is a pleasure to thank Prof. J. Cuntz for his friendly hospitality and for the very stimulating atmosphere of his team.