Bachir Bekka

activity

20052025

most citedOperator-algebraic superrigidity for

S L_{n} (Z), n \geq 3

47 citations · 76 across the 21 of their papers we have counts for

collaborators

28 papers

cond-mat.mtrl-sci2025

Representations of the symmetry groups of infinite crystals

Bachir Bekka, Christian Brouder

We investigate the representations of the symmetry groups of infinite crystals. Crystal symmetries are usually described as the finite symmetry group of a finite crystal with perio…

math.RT2025

Canonical Commutation Relations: A quick proof of the Stone-von Neumann theorem and an extension to general rings

Bachir Bekka

Let $R$ be a (not necessary commutative) ring with unit, $d \geq 1$ an integer, and $λ$ a unitary character of the additive group $(R, +) .$ A pair $(U, V)$ of unitary representations…

math.RT2024

The $L_{p}$ -dual space of a semisimple Lie group

Bachir Bekka

Let $G$ be a semisimple Lie group. We describe the irreducible representations of $G$ by linear isometries on $L_{p}$ -spaces for $p \in (1, + \infty)$ with $p \neq = 2.$ More precisely, we…

math.GR2023

On Bohr compactifications and profinite completions of group extensions

Bachir Bekka

Let $G = N ⋊ H$ be a locally compact group which is a semi-direct product of a closed normal subgroup $N$ and a closed subgroup $H .$ The Bohr compactification $Bohr (G)$ a…

math.GR2023★ 1 cited

The Bohr compactification of an arithmetic group

Bachir Bekka

Given a group $Γ,$ its Bohr compactification $Bohr (Γ)$ and its profinite completion $Prof (Γ)$ are compact groups naturally associated to $Γ$ ; moreover…

math.DS2021★ 1 cited

On the spectral theory of groups of automorphisms of $S$ -adic nilmanifolds

Bachir Bekka, Yves Guivarc'h

Let $S = {p_{1}, \dots, p_{r}, \infty}$ for prime integers $p_{1}, \dots, p_{r} .$ Let $X$ be an $S$ -adic compact nilmanifold, equipped with the unique translation invariant probability meas…

math.OA2020

The Plancherel formula for countable groups

Bachir Bekka

We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group $Γ$ into a direct integral of factor re…

math.RT2020

On unitary representations of algebraic groups over local fields

Bachir Bekka, Siegfried Echterhoff

Let $G$ be an algebraic group over a local field $k$ of characteristic zero. We show that the locally compact group $G (k)$ consisting of the $\math…

math.GR2020

Characters of algebraic groups over number fields

Bachir Bekka, Camille Francini

Let $k$ be a number field, $G$ an algebraic group defined over $k$ , and $G (k)$ the group of $k$ -rational points in $G .$ We determine the set of functions…

math.GR2019

Unitary representations of groups, duals, and characters

Bachir Bekka, Pierre de la Harpe

This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most i…

math.GR2019

Character rigidity of simple algebraic groups

Bachir Bekka

We prove the following extension of Tits' simplicity theorem. Let $k$ be an infinite field, $G$ an algebraic group defined and quasi-simple over $k,$ and $G (k)$ the group of $k$ -ra…

math.GR2019★ 10 cited

Quasi-regular representations of discrete groups and associated C*-algebras

Bachir Bekka, Mehrdad Kalantar

Let $G$ be a countable group. We introduce several equivalence relations on the set $Sub (G)$ of subgroups of $G$ , defined by properties of the quasi-regular representations $…

math.OA2018

Infinite characters on $G L_{n} (Q)$ , on $S L_{n} (Z),$ and on groups acting on trees

Bachir Bekka

Answering a question of J. Rosenberg, we construct the first examples of infinite characters on $G L_{n} (K)$ for a global field $K$ and $n \geq 2.$ The case $n = 2$ is…

math.OA2018

Property (T) for locally compact groups and C*-algebras

Bachir Bekka, Chi-Keung Ng

Let $G$ be a locally compact group and let $C^{*} (G)$ and $C_{r}^{*} (G)$ be the full group $C^{*}$ -algebra and the reduced group $C^{*}$ -algebra of $G$ . We investigate the relationship betwe…

math.DS2018

Spectral gap property and strong ergodicity for groups of affine transformations of solenoids

Bachir Bekka, Camille Francini

Let X be a solenoid, that is, a compact finite dimensional connected abelian group with normalized Haar measure m, and let G be a countable discrete group acting on X by continuous…

math.GR2016★ 1 cited

Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions

Bachir Bekka

Let $G$ be a compactly generated locally compact group and $(π, H)$ a unitary representation of $G .$ The $1$ -cocycles with coefficients in $π$ which are harmonic (with res…

math.GR2016★ 4 cited

Spectral rigidity of group actions on homogeneous spaces

Bachir Bekka

Actions of a locally compact group G on a measure space X give rise to unitary representations of G on Hilbert spaces. We review results on the rigidity of these actions from the s…

math.OA2015

Local rigidity for actions of Kazhdan groups on non commutative $L_{p}$ -spaces

Bachir Bekka

Given a discrete group $Γ$ , a finite factor $N$ and a real number $p \in [1, + \infty)$ with $p \neq = 2,$ we are concerned with the rigidity of actions of $Γ$ by linear isomet…

math.GR2014

Irreducible affine isometric actions on Hilbert spaces

Bachir Bekka, Thibault Pillon, Alain Valette

We undertake a systematic study of irreducible affine isometric actions of locally compact groups on Hilbert spaces. It turns out that, while that are a few parallels of this study…

math.GR2013

On groups with Property (T_lp)

Bachir Bekka, Baptiste Olivier

Let p be a real number with 1<p and different from 2. We study Property (T_lp) for a second countable locally compact group G. Property (T_lp) is a weak version of Kazhdan's Proper…

math.DS2011

On the spectral theory of groups of affine transformations of compact nilmanifolds

Bachir Bekka, Yves Guivarc'h

Let $N$ be a connected and simply connected nilpotent Lie group, $Λ$ a lattice in $N$ , and $X = N /Λ$ the corresponding nilmanifold. Let $A f f (X)$ be the group of affine transformation…

math.GR2010

Groups with faithful irreducible projective unitary representations

Bachir Bekka, Pierre de la Harpe

For a countable group G and a multiplier c on G with values in the circle, we study the property of G having a unitary projective c-representation which is both irreducible and pro…

math.DS2009

Lattices with and lattices without spectral gap

Bachir Bekka, Alexander Lubotzky

The following two results are shown. 1) Let $G$ be the $k$ -rational points of a simple algebraic group over a local field $k$ and let $H$ be a lattice in $G .$ Then the regular repr…

math.GR2009★ 9 cited

A spectral gap property for subgroups of finite covolume in Lie groups

Bachir Bekka, Yves Cornulier

Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation lambda_{G/H} of G on L^2(G/H) has a…

Representations of the symmetry groups of infinite crystals

Canonical Commutation Relations: A quick proof of the Stone-von Neumann theorem and an extension to general rings

The Lp​-dual space of a semisimple Lie group

On Bohr compactifications and profinite completions of group extensions

The Bohr compactification of an arithmetic group

On the spectral theory of groups of automorphisms of S-adic nilmanifolds

The Plancherel formula for countable groups

On unitary representations of algebraic groups over local fields

Characters of algebraic groups over number fields

Unitary representations of groups, duals, and characters

Character rigidity of simple algebraic groups

Quasi-regular representations of discrete groups and associated C*-algebras

Infinite characters on GLn​(Q), on SLn​(Z), and on groups acting on trees

Property (T) for locally compact groups and C*-algebras

Spectral gap property and strong ergodicity for groups of affine transformations of solenoids

Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions

Spectral rigidity of group actions on homogeneous spaces

Local rigidity for actions of Kazhdan groups on non commutative Lp​-spaces

Irreducible affine isometric actions on Hilbert spaces

On groups with Property (T_lp)

On the spectral theory of groups of affine transformations of compact nilmanifolds

Groups with faithful irreducible projective unitary representations

Lattices with and lattices without spectral gap

A spectral gap property for subgroups of finite covolume in Lie groups

The $L_{p}$ -dual space of a semisimple Lie group

On the spectral theory of groups of automorphisms of $S$ -adic nilmanifolds

Infinite characters on $G L_{n} (Q)$ , on $S L_{n} (Z),$ and on groups acting on trees

Local rigidity for actions of Kazhdan groups on non commutative $L_{p}$ -spaces