arXiv subjects

Bachir Bekka

Publications and source records attributed to Bachir Bekka.

At least 19 recordsLinked to original sources

arXiv2025

Representations of the symmetry groups of infinite crystals

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 periodic boundary conditions, for which the Brillouin zone is a finite set of points. However, to deal with the continuous crystal momentum $\mathbf{k}$ required to discuss the continuity, singularity or analyticity of band energies $ε_n(\mathbf{k})$ and Bloch states $ψ_{\mathbf{k}}$, we need to consider infinite crystals. The symmetry groups of infinite crystals belong to the category of infinite non-compact groups, for which many standard tools of group theory break down. For example, character theory is no longer available for these groups and we use harmonic analysis to build the group algebra, the regular representation, the induction of irreducible representations of the crystallographic group from projective representations of the point groups and the decomposition of a representation into its irreducible parts. We deal with magnetic and non-magnetic groups in arbitrary dimensions. In the last part of the paper, we discuss Mackey's restriction of an induced representation to a subgroup, the tensor product of induced representations and the symmetric and antisymmetric squares of induced representations.

cond-mat.mtrl-sci↗

arXiv2025

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

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 $U$ and $V$ of $R^d$ on a Hilbert space $\mathcal{H}$ is said to satisfy the canonical commutation relations (relative to $λ$) if $U(a) V(b)= λ(a\cdot b)V(b) U(a)$ for all $a=(a_1, \dots, a_d), b= (b_1, \dots, b_d)\in R^d$, where $a\cdot b= \sum_{k=1}^d a_k b_k.$ We give a new and quick proof of the classical Stone von Neumann Theorem about the essential uniqueness of such a pair in the case where $R$ is a local field (e.g. $R= \mathbf{R}$). Our methods allow us to give the following extension of this result to a general locally compact ring $R$. For a unitary representation $U$ of $R^d$ on a Hilbert space $\mathcal{H}, $ define the inflation $U^{(\infty)}$ of $U$ as the (countably) infinite multiple of $U$ on $\mathcal{H}^{(\infty)}=\oplus_{i\in \mathbf{N}} \mathcal{H}$. Let $(U_1, V_1), (U_2, V_2)$ be two pairs of unitary representations of $R^d$ on corresponding Hilbert spaces $\mathcal{H}_1, \mathcal{H}_2$ satisfying the canonical commutation relations (relative to $λ$). Provided that $λ$ satisfies a mild faithful condition, we show that the inflations $(U_1^{(\infty)}, V_1^{(\infty)}), (U_2^{(\infty)}, V_2^{(\infty)})$ are approximately equivalent, that is, there exists a sequence $(Φ_n)_n$ of unitary isomorphisms $Φ_n: \mathcal{H}_1^{(\infty)}\to \mathcal{H}_2^{(\infty)}$ such that $\lim_{n} \Vert U_2^{(\infty)}(a) - Φ_n U_1^{(\infty)}(a) Φ_n^{*}\Vert=0$ and $\lim_{n} \Vert V_2^{(\infty)}(b) - Φ_n V_1^{(\infty)}(b) Φ_n^{*}\Vert=0,$ uniformly on compact subsets of $R^d.$

math.RT↗

arXiv2024

The $L_p$-dual space of a semisimple Lie group

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 show that, for every such representation $π,$ there exists a parabolic subgroup $Q$ of $G$ such that $π$ is equivalent to the natural representation of $G$ on $L_p(G/Q)$ twisted by a unitary character of $Q.$ When $G$ is of real rank one, we give a complete classification of the possible irreducible representations of $G$ on an $L_p$-space for $p\neq 2,$ up to equivalence.

math.RT↗

arXiv2023

On Bohr compactifications and profinite completions of group extensions

Let $G= N\rtimes H$ 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 ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of $G$ are, respectively, isomorphic to semi-direct products $ Q_1 \rtimes {\rm Bohr}(H)$ and $ Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of $H$ on appropriate subsets of the dual space of $N$. In the case where $N$ is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where $A$ is the group of unitary characters of $N$ with finite $H$-orbits and $B$ the subgroup of $A$ of characters with finite image. Necessary and sufficient conditions are deduced for $G$ to be maximally almost periodic or residually finite. We apply the results to the case where $G= Λ\wr H$ is a wreath product of countable groups; we show in particular that ${\rm Bohr}(Λ\wr H)$ is isomorphic to ${\rm Bohr}(Λ^{\rm Ab}\wr H)$ and ${\rm Prof}(Λ\wr H)$ is isomorphic to ${\rm Prof}(Λ^{\rm Ab} \wr H),$ where $Λ^{\rm Ab}=Λ/ [Λ, Λ]$ is the abelianization of $Λ.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when $G$ is a lamplighter group and when $G$ is the Heisenberg group over a unital commutative ring.

math.GR↗

arXiv2023

The Bohr compactification of an arithmetic group

Given a group $Γ,$ its Bohr compactification $\operatorname{Bohr}(Γ)$ and its profinite completion $\operatorname{Prof}(Γ)$ are compact groups naturally associated to $Γ$; moreover, $\operatorname{Prof}(Γ)$ can be identified with the quotient of $\operatorname{Bohr}(Γ)$ by its connected component $\operatorname{Bohr}(Γ)_0.$ We study the structure of $\operatorname{Bohr}(Γ)$ for an arithmetic subgroup $Γ$ of an algebraic group $G$ over $\mathbf{Q}$. When $G$ is unipotent, we show that $\operatorname{Bohr}(Γ)$ can be identified with the direct product $\operatorname{Bohr}(Γ^{\rm Ab})_0\times \operatorname{Prof}(Γ)$, where $Γ^{\rm Ab}= Γ/[Γ, Γ]$ is the abelianization of $Γ.$ In the general case, using a Levi decomposition $G= U\rtimes H$ (where $U$ is unipotent and $H$ is reductive), we show that $\operatorname{Bohr}(Γ)$ can be described as the semi-direct product of a certain quotient of $\operatorname{Bohr}(Γ\cap U)$ with $\operatorname{Bohr}(Γ\cap H)$. When $G$ is simple and has higher $\mathbf{R}$-rank, $\operatorname{Bohr}(Γ)$ is isomorphic, up to a finite group, to the product $K\times \operatorname{Prof}(Γ),$ where $K$ is the maximal compact factor of the real Lie group $G(\mathbf{R}).$

math.GR↗

arXiv2021

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

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 measure $μ.$ We characterize the countable groups $Γ$ of automorphisms of $X$ for which the Koopman representation $κ$ on $L^2(X,μ)$ has a spectral gap. More specifically, we show that $κ$ does not have a spectral gap if and only if there exists a non-trivial $Γ$-invariant quotient solenoid (that is, a finite-dimensional, connected, compact abelian group) on which $Γ$ acts as a virtually abelian group.

math.DS↗

arXiv2020

The Plancherel formula for countable groups

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 representations. Our main result gives a precise description of this decomposition in terms of the Plancherel formula of the FC-center $Γ_{\rm fc}$ of $Γ$ (that is, the normal sugbroup of $Γ$ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of $Γ_{\rm fc}$. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma's and Kaniuth's theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II.

math.OA↗

arXiv2020

Character rigidity of simple algebraic groups

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 $k$-rational points of $G.$ Let $G(k)^+$ be the subgroup of $G(k)$ generated by the unipotent radicals of parabolic subgroups of $G$ defined over $k$ and $PG(k)^+$ the quotient of $G(k)^+$ by its center. Then every normalized function of positive type on $PG(k)^+$ which is constant on conjugacy classes is a convex combination of $1_{PG(k)^+}$ and $δ_e.$ As corollary, we obtain that the only ergodic invariant random subgroups (IRS) of $PG(k)^+$ are $δ_{PG(k)^+}$ and $δ_{\{e\}},$ when $k$ is countable. A further consequence is that, when $k$ is a global field and $G$ is $k$-isotropic and has trivial center, every measure preserving ergodic action of $G(k)$ on a probability space either factorizes through the abelianization of $G(k)$ or is essentially free.

math.GR↗

arXiv2020

On unitary representations of algebraic groups over local fields

Let $\mathbf{G}$ be an algebraic group over a local field $\mathbf k$ of characteristic zero. We show that the locally compact group $\mathbf G(\mathbf k)$ consisting of the $\mathbf k$-rational points of $\mathbf G$ is of type I. Moreover, we complete Lipsman's characterization of the groups $\mathbf G$ for which every irreducible unitary representation of $\mathbf G(\mathbf k)$ is a CCR representation and show at the same time that such groups $\mathbf G(\mathbf k)$ are trace class as studied recently by Deitmar and van Dijk.

math.RT↗

arXiv2020

Characters of algebraic groups over number fields

Let $k$ be a number field, $\mathbf{G}$ an algebraic group defined over $k$, and $\mathbf{G}(k)$ the group of $k$-rational points in $\mathbf{G}.$ We determine the set of functions on $\mathbf{G}(k)$ which are of positive type and conjugation invariant, under the assumption that $\mathbf{G}(k)$ is generated by its unipotent elements. An essential step in the proof is the classification of the $\mathbf{G}(k)$-invariant ergodic probability measures on an adelic solenoid naturally associated to $\mathbf{G}(k);$ this last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of $S$-adic Lie groups.

math.GR↗

arXiv2019

Unitary representations of groups, duals, and characters

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 important notions are defined for topological groups, but a special attention is paid to the case of discrete groups. The unitary dual of a group $G$ is the space of equivalence classes of its irreducible unitary representations; it is both a topological space and a Borel space. The primitive dual is the space of weak equivalence classes of unitary irreducible representations. The normal quasi-dual is the space of quasi-equivalence classes of traceable factor representations; it is parametrized by characters, which can be finite or infinite. The theory is systematically illustrated by a series of specific examples: Heisenberg groups, affine groups of infinite fields, solvable Baumslag-Solitar groups, lamplighter groups, and general linear groups. Operator algebras play an important role in the exposition, in particular the von Neumann algebras associated to a unitary representation and C*-algebras associated to a locally compact group.

math.GR↗

arXiv2019

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

Let $G$ be a countable group. We introduce several equivalence relations on the set ${\rm Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations $λ_{G/H}$ associated to $H\in {\rm Sub}(G)$ and compare them to the relation of $G$-conjugacy of subgroups. We define a class ${\rm Sub}_{\rm sg}(G)$ of subgroups (these are subgroups with a certain spectral gap property) and show that they are rigid, in the sense that the equivalence class of $H\in {\rm Sub}_{\rm sg}(G)$ for any one of the above equivalence relations coincides with the $G$-conjugacy class of $H$. Next, we introduce a second class ${\rm Sub}_{\rm w-par}(G)$ of subgroups (these are subgroups which are weakly parabolic in some sense) and we establish results concerning the ideal structure of the $C^*$-algebra $C^*_{λ_{G/H}}(G)$ generated by $λ_{G/H}$ for subgroups $H$ which belong to either one of the classes ${\rm Sub}_{\rm w-par}(G)$ and ${\rm Sub}_{\rm sg}(G)$. Our results are valid, more generally, for induced representations ${\rm Ind}_H^G σ$, where $σ$ is a representation of $H\in {\rm Sub}(G)$.

math.GR↗

arXiv2018

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

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 affine transformations. We show that the probability measure preserving action of G on (X,m) does not have the spectral gap property if and only if there exists a p(G)-invariant proper subsolenoid Y of X such that the image of G in the affine group Aff(X/Y) of X/Y is a virtually solvable group, where p(G) is the automorphism part of G. When G is finitely generated or when X is a p-adic solenoid, the subsolenoid Y can be chosen so that the image of G in Aff(X/Y) is virtually abelian. In particular, an action of a group by affine transformations on a solenoid has the spectral gap property if and only if this action is strongly ergodic.

math.DS↗

arXiv2018

Infinite characters on $GL_n(\mathbf{Q})$, on $SL_n(\mathbf{Z}),$ and on groups acting on trees

Answering a question of J. Rosenberg, we construct the first examples of infinite characters on $GL_n(\mathbf{K})$ for a global field $\mathbf{K}$ and $n\geq 2.$ The case $n=2$ is deduced from the following more general result. Let $G$ a non amenable countable subgroup acting on locally finite tree $X$. Assume either that the stabilizer in $G$ of every vertex of $X$ is finite or that the closure of the image of $G$ in ${\rm Aut}(X)$ is not amenable. We show that $G$ has uncountably many infinite dimensional irreducible unitary representations $(π, \mathcal{H})$ of $G$ which are traceable, that is, such that the $C^*$-subalgebra of $\mathcal{B}(\mathcal{H})$ generated by $π(G)$ contains the algebra of the compact operators on $\mathcal{H}.$ In the case $n\geq 3,$ we prove the existence of infinitely many characters for $G=SL_n(R)$, where $n\geq 3$ and $R$ is an integral domain such that $G$ is not amenable. In particular, the group $SL_n(\mathbf{Z})$ has infinitely many such characters for $n\geq 2.$

math.OA↗

arXiv2018

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

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 between Property $(T)$ for $G$ and Property $(T)$ as well as its strong version for $C^*(G)$ and $C^*_r(G)$. We show that $G$ has Property $(T)$ if (and only if) $C^*(G)$ has Property $(T)$. In the case where $G$ is a locally compact IN-group, we prove that $G$ has Property $(T)$ if and only if $C^*_r(G)$ has strong Property $(T)$. We also show that $C^*_r(G)$ has strong Property $(T)$ for every non-amenable locally compact group $G$ for which $C^*_r(G)$ is nuclear. Some of these groups (as for instance $G=SL_2(\mathbf{R})$) do not have Property $T$.

math.OA↗

arXiv2016

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

Let $G$ be a compactly generated locally compact group and $(π, \mathcal H)$ a unitary representation of $G.$ The $1$-cocycles with coefficients in $π$ which are harmonic (with respect to a suitable probability measure on $G$) represent classes in the first reduced cohomology $\bar{H}^1(G,π).$ We show that harmonic $1$-cocycles are characterized inside their reduced cohomology class by the fact that they span a minimal closed subspace of $\mathcal H.$ In particular, the affine isometric action given by a harmonic cocycle $b$ is irreducible (in the sense that $\mathcal H$ contains no non-empty, proper closed invariant affine subspace) if the linear span of $b(G)$ is dense in $\mathcal H.$ The converse statement is true, if $π$ moreover has no almost invariant vectors. Our approach exploits the natural structure of the space of harmonic $1$-cocycles with coefficients in $π$ as a Hilbert module over the von Neumann algebra $π(G)',$ which is the commutant of $π(G)$. Using operator algebras techniques, such as the von Neumann dimension, we give a necessary and sufficient condition for a factorial representation $π$ without almost invariant vectors to admit an irreducible affine action with $π$ as linear part.

math.GR↗

arXiv2016

Spectral rigidity of group actions on homogeneous spaces

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 spectral point of view, that is, results about the existence of a spectral gap for associated averaging operators and their consequences. We will deal both with spaces X with an infinite measure as well as with spaces with an invariant probability measure. The spectral gap property has several striking applications to group theory, geometry, ergodic theory, operator algebras, graph theory, theoretical computer science, etc.

math.GR↗

arXiv2015

Local rigidity for actions of Kazhdan groups on non commutative $L_p$-spaces

Given a discrete group $Γ$, a finite factor $\mathcal N$ and a real number $p\in [1, +\infty)$ with $p\neq 2,$ we are concerned with the rigidity of actions of $Γ$ by linear isometries on the $L_p$-spaces $L_p(\mathcal N)$ associated to $\mathcal N$. More precisely, we show that, when $Γ$ and $\mathcal N$ have both Property (T) and under some natural ergodicity condition, such an action $π$ is locally rigid in the group $G$ of linear isometries of $L_p(\mathcal N)$, that is, every sufficiently small perturbation of $π$ is conjugate to $π$ under $G$. As a consequence, when $Γ$ is an ICC Kazhdan group, the action of $Γ$ on its von Neumann algebra ${\mathcal N}(Γ)$, given by conjugation, is locally rigid in the isometry group of $L_p({\mathcal N}(Γ)).$

math.OA↗