Publications (11)11.87 Total impact
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ABSTRACT: For a compact quantum group $\mathbb G$ of Kac type, we study the existence of a Haar tracepreserving 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 $O_N^+$.  [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_1factors and Brannan's result showing that the orthogonal free quantum groups have Haagerup's approximation property. We also deduce OzawaPopa's property strong (HH) and give a new proof of Isono's result about strong solidity.  [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 L2L2Betti number of Ao(In)Ao(In). 
Article: The Ktheory of free quantum groups
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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 PimsnerVoiculescu 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 BaumConnes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the BaumConnes 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 Diracdual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the BaumConnes conjecture to our setting.  [Show abstract] [Hide abstract]
ABSTRACT: The halfliberated 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 twistingtype 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 ClebschGordan 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 Ktheory of the reduced group C algebras carry over to the quantum case.  [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  [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 AkemannOstrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.  [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 noninvolutivity of the edgereversing operator and vanishes in the classical case. We end with applications to Property AO and Ktheory.  [Show abstract] [Hide abstract]
ABSTRACT: The basic notions and results of equivariant KKtheory 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 BassSerre tree, we establish the Kamenability of amalgamated free products of amenable discrete quantum groups.
Publication Stats
203  Citations  
11.87  Total Impact Points  
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Institutions

20062014

Université de Caen Normandie
 • Laboratoire de Mathématiques Nicolas Oresme
 • Département de Mathématiques
Caen, Lower Normandy, France


2005

University of Münster
 Institute of Mathematics
Muenster, North RhineWestphalia, Germany
