arXiv subjects

Bachir Bekka

Publications and source records attributed to Bachir Bekka.

28 records · Page 2Linked to original sources

arXiv2014

Irreducible affine isometric actions on Hilbert spaces

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 to the by now classical theory of irreducible unitary representations, these two theories differ in several aspects (for instance, the direct sum of two irreducible affine actions can still be irreducible). One of the main tools we use is an affine version of Schur's lemma characterizing the irreducibility of an affine isometric group action. This enables us to describe for instance the irreducible affine isometric actions of nilpotent groups. As another application, a short proof is provided for the following result of Neretin: the restriction to a cocompact lattice of an irreducible affine action of locally compact group remains irreducible. We give a necessary and sufficient condition for a fixed unitary representation to be the linear part of an irreducible affine action. In particular, when the unitary representation is a multiple of the regular representation of a discrete group G, we show how this question is related to the L2-Betti number of G. After giving a necessary and sufficient condition for a direct sum of irreducible affine actions to be irreducible, we show the following super-rigidity result: if G is product of two or more locally compact groups, then every irreducible affine action of any irreducible co-compact lattice in G extends to an affine action of G, provided the linear part of this action does not weakly contain the trivial representation.

math.GR↗

arXiv2013

On groups with Property (T_lp)

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 Property (T), defined in terms of the orthogonal representations of G on the sequence space lp. We show that Property (T_lp) for a totally disconnected group G is characterized by an isolation property of the trivial representation from the quasi-regular representations associated to open subgroups of G. Groups with Property (T_lp) share some important properties with Kazhdan groups (compact generation, compact abelianization, ...). Simple algebraic groups over non-archimedean local fields as well as automorphism groups of regular trees have Property (T_lp). In the case of discrete groups, Property (T_lp) implies Lubotzky's Property tau and is implied by Property (F) of Glasner and Monod. We show that an irreducible lattice in a product of two locally compact groups G and H have Property (T_lp), whenever G has Property (T) and H is connected and minimally almost periodic.

math.GR↗

arXiv2011

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

Let $N$ be a connected and simply connected nilpotent Lie group, $Λ$ a lattice in $N$, and $X=N/Λ$ the corresponding nilmanifold. Let $Aff(X)$ be the group of affine transformations of $X$. We characterize the countable subgroups $H$ of $Aff(X)$ for which the action of $H$ on $X$ has a spectral gap, that is, such that the associated unitary representation $U$ of $H$ on the space of functions from $L^2(X)$ with zero mean does not weakly contain the trivial representation. Denote by $T$ the maximal torus factor associated to $X$. We show that the action of $H$ on $X$ has a spectral gap if and only if there exists no proper $H$-invariant subtorus $S$ of $T$ such that the projection of $H$ on $Aut (T/S)$ has an abelian subgroup of finite index. We first establish the result in the case where $X$ is a torus. In the case of a general nilmanifold, we study the asymptotic behaviour of matrix coefficients of $U$ using decay properties of metaplectic representations of symplectic groups. The result shows that the existence of a spectral gap for subgroups of $Aff(X)$ is equivalent to strong ergodicity in the sense of K.Schmidt. Moreover, we show that the action of $H$ on $X$ is ergodic (or strongly mixing) if and only if the corresponding action of $H$ on $T$ is ergodic (or strongly mixing).

math.DS↗

arXiv2010

Groups with faithful irreducible projective unitary representations

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 projectively faithful. We show that this property is equivalent to G being the quotient of an appropriate group by its centre. A criterion is given in terms of the minisocle of G. Several examples are described to show the existence of various behaviours.

math.GR↗

arXiv2009

Lattices with and lattices without spectral gap

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 representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are no almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with zero mean). 2) There exist locally compact simple groups $G$ and lattices $H$ for which $L^2(G/H)$ has no spectral gap. This answers in the negative a question asked by Margulis. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of a $k$-regular tree for $k>2.$

math.DS↗

arXiv2009

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

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 spectral gap, that is, the restriction of lambda_{G/H} to the orthogonal of the constants in L^2(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

math.GR↗

arXiv2008

Irreducibly represented groups

A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented. We indicate some consequences of this for operator algebras. In particular, we charaterise up to isomorphism the countable subgroups $Δ$ of the unitary group of a separable infinite dimensional Hilbert space $\Cal H$ of which the bicommutants $Δ''$ (in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on $\Cal H$.

math.GR↗

arXiv2006

Operator-algebraic superrigidity for $SL_n(\mathbb Z),n\geq 3$

For $n\geq 3,$ let $Γ=SL_n(\mathbb Z).$ We prove the following superridigity result for $Γ$ in the context of operator algebras. Let $L(Γ)$ be the von Neumann algebra generated by the left regular representation of $Γ.$ Let $M$ be a finite factor and let $U(M)$ be its unitary group. Let $π: Γ\to U(M)$ be a group homomorphism such that $π(Γ)''=M.$ Then \begin{itemize} \item[(i)] either $M$ is finite dimensional, or \item [(ii)] there exists a subgroup of finite index $Λ$ of $Γ$ such that $π|_Λ$ extends to a homomorphism $U(L(Λ))\to U(M).$ \end{itemize} The result is deduced from a complete description of the tracial states on the full $C^*$--algebra of $Γ.$ As another application, we show that the full $C^*$--algebra of $Γ$ has no faithful tracial state.

math.OA↗

arXiv2005

A spectral gap property for random walks under unitary representations

Let $G$ be a locally compact group and $μ$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(π, \cal H)$ of $G,$ we study spectral properties of the operator $π(μ)$ acting on $\cal H.$ Assume that $μ$ is adapted and that the trivial representation $1_G$ is not weakly contained in the tensor product $π\otimes \barπ.$ We show that $π(μ)$ has a spectral gap, that is, for the spectral radius $r_{\rm spec}(π(μ))$ of $π(μ),$ we have $r_{\rm spec}(π(μ))<1.$ This provides a common generalization of several previously known results. Another consequence is that, if $G$ has Kazhdan's Property (T), then $r_{\rm spec}(π(μ))<1$ for every unitary representation $π$ of $G$ without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.

math.DS↗

arXiv2005

A property (T) for C*-algebras

We define a notion of Property (T) for an arbitrary $C^*$-algebra $A$ admitting a tracial state. We extend this to a notion of Property (T) for the pair $(A,B),$ where $B$ is a $C^*$-subalgebra of $A.$ Let $Γ$ be a discrete group and $C^*_r(Γ)$ its reduced algebra. We show that $C^*_r(Γ)$ has Property (T) if and only if the group $Γ$ has Property (T) . More generally, given a subgroup $Λ$ of $Γ$, the pair $(C^*_r(Γ),C^*_r(Λ))$ has Property (T) if and only if the pair of groups $(Γ, Λ)$ has Property (T).

math.OA↗