arXiv subjects
Publications and source records attributed to François Lemeux.
In this paper we study two semigroups of completely positive unital self-adjoint maps on the von Neumann algebras of the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that these semigroups satisfy ultracontractivity and hypercontractivity estimates. We also give results regarding spectral gap and logarithmic Sobolev inequalities.
In this paper, we find the fusion rules for the free wreath product quantum groups $\mathbb{G}\wr_*S_N^+$ for all compact matrix quantum groups of Kac type $\mathbb{G}$ and $N\ge4$. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of $\mathbb{G}\wr_*S_N^+$. The combinatorial properties of the intertwiner spaces in $\mathbb{G}\wr_*S_N^+$ then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between $\mathbb{G}\wr_*S_N^+$ and a compact quantum group whose dual is a discrete quantum subgroup of the free product $\widehat{\mathbb{G}}*\widehat{SU_q(2)}$, for some $0<q\le1$. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as Haagerup property, weak amenability and exactness.
In this paper we find the fusion rules of the free wreath products $\widehatΓ\wr_*S_N^+$ for any (discrete) group $Γ$. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations. We then apply the knowledge of the fusion rules to prove, in most cases, several operator algebraic properties of the associated reduced $C^*$-algebras such as simplicity and uniqueness of the trace. We also prove that the associated von Neumann algebra is a full type $II_1$-factor and that the dual of $\widehatΓ\wr_*S_N^+$ has the Haagerup approximation property for all finite groups $Γ$.
In this paper we prove that the duals of the quantum reflection groups have the Haagerup property for all $N\ge4$ and $s\in[1,\infty)$. We use the canonical arrow onto the quantum permutation groups, and we describe how the characters of the quantum reflection groups behave with respect to this canonical morphism thanks to the description of the fusion rules binding irreducible corepresentations of $H_N^{s+}$. This allows us to construct states on the central algebra of $H_{N}^{s+}$ and to use a fundamental theorem proved by M.Brannan giving a method to construct nets of trace-preserving, normal, unital and completely positive maps on the von Neumann algebra of a compact quantum group of Kac type.