[Show abstract][Hide abstract] ABSTRACT: For a compact quantum group $\mathbb G$ of Kac type, we study the existence
of a Haar trace-preserving embedding of the von Neumann algebra
$L^\infty(\mathbb G)$ into an ultrapower of the hyperfinite II$_1$-factor (the
Connes embedding property for $L^\infty(\mathbb G)$). We establish a connection
between the Connes embedding property for $L^\infty(\mathbb G)$ and the
structure of certain quantum subgroups of $\mathbb G$, and use this to prove
that the II$_1$-factors $L^\infty(O_N^+)$ and $L^\infty(U_N^+)$ associated to
the free orthogonal and free unitary quantum groups have the Connes embedding
property for all $N \ge 4$. As an application, we deduce that the free entropy
dimension of the standard generators of $L^\infty(O_N^+)$ equals $1$ for all $N
\ge 4$. We also mention an application of our work to the problem of
classifying the quantum subgroups of $O_N^+$.
Transactions of the American Mathematical Society 12/2014; DOI:10.1090/tran/6752 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that the orthogonal free quantum groups are not inner amenable and we
construct an explicit proper cocycle weakly contained in the regular
representation. This strengthens the result of Vaes and the second author,
showing that the associated von Neumann algebras are full II_1-factors and
Brannan's result showing that the orthogonal free quantum groups have
Haagerup's approximation property. We also deduce Ozawa-Popa's property strong
(HH) and give a new proof of Isono's result about strong solidity.
International Mathematics Research Notices 02/2014; DOI:10.1093/imrn/rnu268 · 1.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study existence, uniqueness and triviality of path cocycles in the quantum Cayley graph of universal discrete quantum groups. In the orthogonal case we find that the unique path cocycle is trivial, in contrast with the case of free groups where it is proper. In the unitary case it is neither bounded nor proper. From this geometrical result we deduce the vanishing of the first L2L2-Betti number of Ao(In)Ao(In).
Advances in Mathematics 03/2012; 229(5):2686–2711. DOI:10.1016/j.aim.2012.01.011 · 1.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we study the $ K $-theory of free quantum groups in the sense
of Wang and Van Daele, more precisely, of free products of free unitary and
free orthogonal quantum groups. We show that these quantum groups are $ K
$-amenable and establish an analogue of the Pimsner-Voiculescu exact sequence.
As a consequence, we obtain in particular an explicit computation of the $ K
$-theory of free quantum groups. Our approach relies on a generalization of
methods from the Baum-Connes conjecture to the framework of discrete quantum
groups. This is based on the categorical reformulation of the Baum-Connes
conjecture developed by Meyer and Nest. As a main result we show that free
quantum groups have a $ \gamma $-element and that $ \gamma = 1 $. As an
important ingredient in the proof we adapt the Dirac-dual Dirac method for
groups acting on trees to the quantum case. We use this to extend some
permanence properties of the Baum-Connes conjecture to our setting.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We find the fusion rules for the quantum analogues of the complex reflection groups $H_n^s=\mathbb Z_s\wr S_n$. The irreducible representations can be indexed by the elements of the free monoid $\mathbb N^{*s}$, and their tensor products are given by formulae which remind the Clebsch-Gordan rules (which appear at $s=1$). Comment: 33 pages
[Show abstract][Hide abstract] ABSTRACT: We introduce the Property of Rapid Decay for discrete quantum groups by equivalent characterizations that generalize the classical ones. We investigate examples, proving in particular the Property of Rapid Decay for unimodular free quantum groups. We finally check that the applications to the K-theory of the reduced group C -algebras carry over to the quantum case.
JOURNAL OF OPERATOR THEORY 03/2007; 2(2). · 0.55 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We discuss the notion of growth for discrete quantum groups, with a number of general considerations, and with some explicit computations. Of particular interest is the quantum analogue of Gromov's estimate regarding polynomial growth: we formulate the precise question, and we verify it for the duals of classical Lie groups. Comment: 17 pages
Infinite Dimensional Analysis Quantum Probability and Related Topics 11/2006; DOI:10.1142/S0219025709003677 · 0.73 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Duke Mathematical Journal 10/2005; 140(1). DOI:10.1215/S0012-7094-07-14012-2 · 1.58 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Journal für die reine und angewandte Mathematik (Crelles Journal) 03/2005; 2005(580):101-138. DOI:10.1515/crll.2005.2005.580.101 · 1.43 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some useful properties of subgroups and amalgamated free products of discrete quantum groups. Using these properties and a quantum analogue of the Bass-Serre tree, we establish the K-amenability of amalgamated free products of amenable discrete quantum groups.