arXiv subjects
Publications and source records attributed to Rémi Boutonnet.
We prove that if $A$ is a non-separable abelian tracial von Neuman algebra then its free powers $A^{*n}, 2\leq n \leq \infty$, are mutually non-isomorphic and with trivial fundamental group, $\mathcal F(A^{*n})=1$, whenever $2\leq n<\infty$. This settles the non-separable version of the free group factor problem.
We complete the study of characters on higher rank semisimple lattices initiated in [BH19,BBHP20], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo-Zimmer structure theorem obtained in [BH19] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis' factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.
We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices $Γ< G$ in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras $\operatorname{L}(Γ) \subset M \subset \operatorname{L}(Γ\curvearrowright G/P)$ sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.
We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.
We construct uncountably many infinite characters of type II for $SL_n(\mathbb{Z})$, $n \geq 2$.
We show that stationary characters on irreducible lattices $Γ< G$ of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $Γ< G$, the left regular representation $λ_Γ$ is weakly contained in any weakly mixing representation $π$. We prove that for any such irreducible lattice $Γ< G$, any uniformly recurrent subgroup (URS) of $Γ$ is finite, answering a question of Glasner-Weiss. We also obtain a new proof of Peterson's character rigidity result for irreducible lattices $Γ< G$. The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.
We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.
We investigate the position of amenable subalgebras in arbitrary amalgamated free product von Neumann algebras $M = M_1 \ast_B M_2$. Our main result states that under natural analytic assumptions, any amenable subalgebra of $M$ that has a large intersection with $M_1$ is actually contained in $M_1$. The proof does not rely on Popa's asymptotic orthogonality property but on the study of non normal conditional expectations.
We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in \cite{BISG15}. This property has applications to strong ergodicity, the Banach-Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group of study here is the group $\text{Isom}(\mathbb{R}^d)$ of orientation-preserving isometries of the euclidean space $\mathbb{R}^d$, for $d \geq 3$. We prove that the translation action of a countable dense subgroup $Γ$ on Isom$(\mathbb R^d)$ has local spectral gap, whenever the translation action of the rotation projection of $Γ$ on $\text{SO}(d)$ has spectral gap. Our proof relies on the amenability of $\text{Isom}(\mathbb{R}^d)$ and on work of Lindenstrauss and Varjú, \cite{LV14}.
We show that Shlyakhtenko's free Araki-Woods factors are strongly solid, meaning that for any diffuse amenable von Neumann subalgebra that is the range of a normal conditional expectation, the normalizer remains amenable. This provides the first class of nonamenable strongly solid type III factors.
We show that any amenable von Neumann subalgebra of any free Araki-Woods factor that is globally invariant under the modular automorphism group of the free quasi-free state is necessarily contained in the almost periodic free summand.
We study actions of locally compact groups on von Neumann factors and the associated crossed-product von Neumann algebras. In the setting of totally disconnected groups we provide sufficient conditions on an action $G\curvearrowright Q$ ensuring that the inclusion $Q \subset Q \rtimes G$ is irreducible and that every intermediate subfactor is of the form $Q \rtimes H$ for a closed subgroup $H<G$. This partially generalizes a result of Izumi-Longo-Popa [ILP98] and Choda [Ch78]. We moreover show that one can not hope to use their strategy for non-discrete groups.
We introduce a novel notion of {\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $Γ\curvearrowright G$, whenever $Γ$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group $G$. This extends to the non-compact setting recent works of Bourgain and Gamburd \cite{BG06,BG10}, and Benoist and de Saxcé \cite{BdS14}. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on $G$. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $Γ$-invariant finitely additive measure defined on all bounded measurable subsets of $G$.
We prove that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II$_1$ factors originally introduced by McDuff \cite{MD69a,MD69b} are such examples. This entails the existence of a continuum of non-elementarily equivalent II$_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.
We prove that for any infinite, maximal amenable subgroup $H$ in a hyperbolic group $G$, the von Neumann subalgebra $LH$ is maximal amenable inside $LG$. It provides many new, explicit examples of maximal amenable subalgebras in II$_1$ factors. We also prove similar maximal amenability results for direct products of relatively hyperbolic groups and orbit equivalence relations arising from measure-preserving actions of such groups.
We provide a general criterion to deduce maximal amenability of von Neumann subalgebras $LΛ\subset LΓ$ arising from amenable subgroups $Λ$ of discrete countable groups $Γ$. The criterion is expressed in terms of $Λ$-invariant measures on some compact $Γ$-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C*-algebra.
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined on a standard measure space and which splits as the free product $\mathcal R = \mathcal R_1 \ast \mathcal R_2$ of recurrent subequivalence relations gives rise to a nonamenable factor $\rL(\mathcal R)$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors $\rL^\infty(X) \rtimes Γ$ arising from nonsingular free ergodic actions $Γ\curvearrowright (X, μ)$ on standard measure spaces of amalgamated groups $Γ= Γ_1 \ast_Σ Γ_2$ over a finite subgroup $Σ$.
We generalize W*-superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If Γ is any ICC group which is w-rigid (i.e. it contains an infinite normal subgroup with the relative property (T)) then any mixing Gaussian action σ of Γ is W*-superrigid. More precisely, if ρ is another free ergodic action of a group Λ such that the crossed-product von Neumann algebras associated with ρ and σ are isomorphic, then Λ and Γ are isomorphic, and the actions ρ and σ are conjugate. We prove a similar statement whenever Γ is a non-amenable ICC product of two infinite groups.