arXiv subjects
Publications and source records attributed to Paul Jolissaint.
The aim of the article is to provide characterizations of the Haagerup property for locally compact, second countable groups in terms of approximations of some non-ergodic invariant states by mixing ones for actions on unital $C^*$-algebras one the one hand, and for pairs of tracial von Neumann algebras by mixing binormal states on the other hand.
Let $n\geq 1$ be an odd integer, set $m=(n-1)/2$ and let $M$ be an $n\times n$ matrix whose coefficients are of the form $M_{i,j}=aij+bi+cj+d$ where $0\leq i,j\leq n-1$. Then we prove that for all squares centered at the central coefficient $M_{m,m}$, the mean value of the square equals $M_{m,m}$.
Let $M$ be a von Neumann algebra which acts in a standard way on the Hilbert space $H$, let $G$ be a subgroup of the group of all $*$-automorphisms of $M$. We prove that $M$ is $G$-finite (in the sense of I. Kovács and Szücs, i.e. the set of all normal, $G$-invariant states on $M$ is separating) if and only if, for every $x\in M$ and all $ξ,η\in H$, the coefficient function $g\mapsto \langle g(x)ξ|η\rangle$ is weakly almost periodic.
We define a new approximation property for tracial von Neumann algebras, called \textit{weakly mixing approximation property} which, for discrete groups and II$_1$ factors, is equivalent to the negation of Kazhdan's property (T).
Let $(M,τ,σ,Γ)$ be a (finite) von Neumann dynamical system and let $N$ be a $Γ$-invariant unital von Neumann subalgebra of $M$. If $V\subset L^2(M)$ is a right $N$-submodule whose projection $p_V$ has finite trace in $< M,e_N>$ and is $Γ$-invariant, then we prove that, for every $ε>0$, one can find a $Γ$-invariant submodule $W\subset V$ which has finite rank and such that $Tr(p_V-p_W)<ε$. Furthermore, we also construct a $σ$-cocycle that gives the action of $Γ$ on a basis of $W$. In particular, this answers a question of T. Austin, T. Eisner and T. Tao.
We propose a proof of the Lagrange Interpolation Formula based on the Chinese Remainder Theorem for arbitrary rings. Even such relationships are known, we think that our viewpoint is worth being published.
The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$ the group $\SL_2(\Z)$ for $n$ even, and $\PSL_2(\Z)$ for $n$ odd. We prove that the action is free and ergodic for every $n\geq 2$, that if $h\in \SL_2(\Z)$ is a hyperbolic element and if $n$ is even, then the action of the subgroup generated by $h$ is still ergodic, but also that, for $n$ odd, no amenable subgroup of $\PSL_2(\Z)$ acts ergodically on $\T^n$. We deduce also that every ergodic sub-equivalence relation $\Rr$ of the orbital equivalence relation $\mathcal{S}_n$ of $G_n$ on $\T^n$ is either amenable or rigid, extending a result by Ioana for $n=2$. This result has the following corollaries: firstly, for $n\geq 2$ even, if $H$ is a maximal amenable subgroup of $\SL_2(\Z)$ containing an hyperbolic matrix, then the associated crossed product II$_1$ factor $L^\infty(\T^n)\rtimes H$ is a maximal Haagerup subalgebra of $L^\infty(\T^n)\rtimes \SL_2(\Z)$; secondly , for every $n$, the fundamental group of $L^\infty(\T^n)\rtimes G_n$ is trivial.
Let $M\subset B(\mathcal H)$ be a von Neumann algebra acting on the Hilbert space $\mathcal H$. We prove that $M$ is finite if and only if, for every $x\in M$ and for all vectors $ξ,η\in\mathcal H$, the coefficient function $u\mapsto \langle uxu^*ξ|η\rangle$ is weakly almost periodic on the topological group $U_M$ of unitaries in $M$ (equipped with the weak or strong operator topology). The main device is the unique invariant mean on the $C^*$-algebra $\operatorname{WAP}(U_M)$ of weakly almost periodic functions on $U_M$. Next, we prove that every coefficient function $u\mapsto \langle uxu^*ξ|η\rangle$ is almost periodic if and only if $M$ is a direct sum of a diffuse, abelian von Neumann algebra and finite-dimensional factors. Incidentally, we prove that if $M$ is a diffuse von Neumann algebra, then its unitary group is minimally almost periodic.
The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $σ$-finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.
The aim of the article is to provide a characterization of Kazhdan's property (T) for locally compact, second countable pairs of groups $H\subset G$ in terms of actions on infinite, $σ$-finite measure spaces. It is inspired by the recent characterization of the Haagerup property by similar actions due to T. Delabie, A. Zumbrunnen and the author.
Let $G$ and $H$ be locally compact, second countable groups. Assume that $G$ acts in a measure class preserving way on a standard probability space $(X,μ)$ such that $L^\infty(X,μ)$ has an invariant mean and that there is a Borel cocycle $α:G\times X\rightarrow H$ which is proper in a suitable, natural sense. We show that if $H$ has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does $G$. We observe that it is the case for a weak form of measure equivalence for pairs of discrete groups.
Let $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss the corresponding relative property gamma for pairs of finite factors $N\subset M$ and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property (T).
Let $G$ be a subgroup of a discrete (countable) group $Γ$. We introduce a notion of relative inner amenability of $G$ in $Γ$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss briefly the corresponding relative property gamma for pairs of finite factors and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property $T$.
For any locally compact group $G$, we show the existence and uniqueness up to quasi-equivalence of a unitary $C_0$-representation $π_0$ of $G$ such that all coefficient functions of $C_0$-representations of $G$ are coefficient functions of $π_0$. The present work, strongly influenced by the work of N. Brown and E. Guentner (which dealt exclusively with discrete groups), leads to new characterizations of the Haagerup property: if $G$ is second countable, then it has that property if and only if the representation $π_0$ induces a *-isomorphism of $C^*(G)$ onto $C^*_{π_0}(G)$. When $G$ is discrete, we also relate the Haagerup property to relative strong mixing properties of the group von Neumann algebra $L(G)$ into finite von Neumann algebras.
We discuss different mixing properties for triples of finite von Neumann algebras $B\subset N\subset M$, and we introduce families of triples of groups $H<K<G$ whose associated von Neumann algebras $L(H)\subset L(K)\subset L(G)$ satisfy $\mathcal{N}_{L(G)}(L(H))"=L(K)$. It turns out that the latter equality is implied by two conditions: the equality $\mathcal{N}_G(H)=K$ and the above mentioned mixing properties. Our families of examples also allow us to exhibit examples of pairs $H<G$ such that $L(\mathcal{N}_G(H))\not=\mathcal{N}_{L(G)}(L(H))"$.
We provide a direct and elementary proof of the equivalence between the weak asymptotic homomorphism property for the pair of group von Neumann algebras $L(H)\subset L(G)$ and the embedding into $H$ of the one sided quasi-normalizer of the pair $H<G$, as it is stated by J. Fang, M. Gao and R. R. Smith in a recent article, and we extend the result to triples of groups $H<K<G$.
We prove that many sequences of positive numbers $(a_n)$ defined by finite linear difference equations $a_{n+k}=c_{k-1}a_{n+k-1}+...+c_0a_n$ with suitable non negative reals coefficients $c_i$ satisfy Bendford's Law on the first digit in many bases $b>2$. Our techniques rely on Perron-Frobenius theory via the companion matrix of the characteristic polynomial of the defining equation.
We present families of pairs of finite von Neumann algebras $A\subset M$ where $A$ is a maximal injective masa in the type $\mathrm{II}_1$ factor $M$ with separable predual. Our results make use of the strong mixing and the asymptotic orthogonality properties of $A$ in $M$ and are borrowed from ideas of S. Popa who proved that if $G$ is a non abelian free group and if $a$ is one of its generators, then the von Neumann algebra generated by $a$ is maximal injective in the factor $L(G)$. Our results apply to pairs $H<G$ where $H$ is an infinite abelian subgroup of a suitable amalgamated product group $G$.