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Several rigidity features of von Neumann algebras

Abstract : The purpose of this dissertation is to put on light rigidity properties of several constructions of von Neumann algebras. These constructions relate group theory and ergodic theory to operator algebras.In Chapter II, we study von Neumann algebras associated with measure-Preserving actions of discrete groups: Gaussian actions. These actions are somehow a generalization of Bernoulli actions. We have two goals in this chapter. The first goal is to use the von Neumann algebra associated with an action as a tool to deduce properties of the initial action (see Corollary II.2.16). The second aim is to prove structural results and classification results for von Neumann algebras associated with Gaussian actions. The most striking rigidity result of the chapter is Theorem II.4.5, which states that in some cases the von Neumann algebra associated with a Gaussian action entirely remembers the action, up to conjugacy. Our results generalize similar results for Bernoulli actions ([KT08,CI10,Io11,IPV13]).In Chapter III, we study amalgamated free products of von Neumann algebras. The content of this chapter is obtained in collaboration with C. Houdayer and S. Raum. We investigate Cartan subalgebras in such amalgamated free products. In particular, we deduce that the free product of two von Neumann algebras is never obtained as a group-Measure space construction of a non-Singular action of a discrete countable group on a measured space.Finally, Chapter IV is concerned with von Neumann algebras associated with hyperbolic groups. The content of this chapter is obtained in collaboration with A. Carderi. We use the geometry of hyperbolic groups to provide new examples of maximal amenable (and yet type I) subalgebras in type II_1 factors.
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Rémi Boutonnet. Several rigidity features of von Neumann algebras. General Mathematics [math.GM]. Ecole normale supérieure de lyon - ENS LYON, 2014. English. ⟨NNT : 2014ENSL0901⟩. ⟨tel-01124349⟩

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