arXiv subjects

Roland Vergnioux

Publications and source records attributed to Roland Vergnioux.

18 recordsLinked to original sources

arXiv2026

$C^*$-simplicity and boundary actions of discrete quantum groups

We introduce and investigate several quantum group dynamical notions for the purpose of studying $C^*$-simplicity of discrete quantum groups via the theory of boundary actions. In particular we define a quantum analogue of Powers' Averaging Property (PAP) and a quantum analogue of strongly faithful actions. We show that our quantum PAP implies $C^*$-simplicity and the uniqueness of $σ$-KMS states, and that the existence of a strongly $C^*$-faithful quantum boundary action also implies $C^*$-simplicity and, in the unimodular case, the quantum PAP. We illustrate these results in the case of the unitary free quantum groups $\mathbb{F} U_F$ by showing that they satisfy the quantum PAP and that they act strongly $C^*$-faithfully on their quantum Gromov boundary. Moreover we prove that this particular action of $\mathbb{F} U_F$ is a quantum boundary action.

math.OA↗

arXiv2025

Maximal amenability of the radial subalgebra in free quantum group factors

We show that the radial MASA in the orthogonal free quantum group algebra L(FO_N) is maximal amenable if N is large enough, using the Asymptotic Orthogonality Property. This relies on a detailed study of the corresponding bimodule, for which we construct in particular a quantum analogue of Rădulescu's basis. As a byproduct we also obtain the value of the Pukánszky invariant for this MASA.

math.OA↗

arXiv2023

Hecke algebras and the Schlichting completion for discrete quantum groups

We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the construction of the Schlichting completion to the quantum setting, thus obtaining locally compact quantum groups with compact open quantum subgroups. We study in detail a class of examples arising from quantum HNN extensions.

math.QA↗

arXiv2021

Property RD and hypercontractivity for orthogonal free quantum groups

We prove that the twisted property RD fails to hold for all non Kac type, non amenable orthogonal free quantum groups. In the Kac case we revisit property RD, proving an analogue of the $L_p-L_2$ non-commutative Khintchine inequality for free groups. As an application, we give new and improved hypercontractivity and ultracontractivity estimates for the generalized heat semigroups on free orthogonal quantum groups, both in the Kac and non Kac cases.

math.OA↗

arXiv2020

Noncommutative Furstenberg boundary

We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.

math.OA↗

arXiv2020

The Property of Rapid Decay for Discrete Quantum Groups

We introduce the Property of Rapid Decay for discrete quantum groups by equivalent characterizations that generalize the classical ones. We then 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.

math.OA↗

arXiv2020

K-amenability for amalgamated free products of amenable discrete quantum groups

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 usefull 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.

math.OA↗

arXiv2020

Orientation of quantum Cayley trees and applications

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.

math.OA↗

arXiv2018

Orthogonal free quantum group factors are strongly 1-bounded

We prove that the orthogonal free quantum group factors $\mathcal{L}(\mathbb{F}O_N)$ are strongly $1$-bounded in the sense of Jung. In particular, they are not isomorphic to free group factors. This result is obtained by establishing a spectral regularity result for the edge reversing operator on the quantum Cayley tree associated to $\mathbb{F}O_N$, and combining this result with a recent free entropy dimension rank theorem of Jung and Shlyakhtenko.

math.OA↗

arXiv2016

The radial MASA in free orthogonal quantum groups

We prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing, and we compute the associated bimodule. The proof relies on new properties of the Jones-Wenzl projections and on an estimate of certain scalar products of coefficients of irreducible representations.

math.OA↗

arXiv2015

A cocycle in the adjoint representation of the orthogonal free quantum groups

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.

math.OA↗

arXiv2015

The Connes embedding property for quantum group von Neumann algebras

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 $O_N^+$.

math.OA↗

arXiv2012

Paths in quantum Cayley trees and L^2-cohomology

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 L^2-Betti number of A_o(I_n).

math.OA↗

arXiv2011

The K-theory of free quantum groups

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 $ γ$-element and that $ γ= 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.

math.OA↗

arXiv2009

Invariants of the half-liberated orthogonal group

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.

math.QA↗

arXiv2008

Growth estimates for discrete quantum groups

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.

math.OA↗

arXiv2008

Fusion rules for quantum reflection groups

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 $s=1$).

math.OA↗

arXiv2006

The boundary of universal discrete quantum groups, exactness and factoriality

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.

math.OA↗