Explicit computations of all finite index bimodules for a family of II_1 factors

We study II_1 factors M and N associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result: every finite index M-N-bimodule (in particular, every isomorphism between M and N) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index M-M-bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II_1 factor. We obtain in particular explicit examples of II_1 factors with trivial fusion algebra, i.e. only having trivial finite index subfactors.Comment: Minor modifications, final versio

Rigidity for von Neumann algebras and their invariants

We give a survey of recent classification results for crossed product von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II_1 factors with uncountable fundamental groups and the construction of W*-superrigid actions where the crossed product entirely remembers the initial group action that it was constructed from.Comment: ICM 2010 Proceedings tex

The unitary implementation of a locally compact quantum group action

In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When alpha is an action of a locally compact quantum group on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion of the fixed point algebra in the algebra N in the crossed product, is a basic construction. When alpha is an outer and integrable action on a factor N we prove that the inclusion of the fixed point algebra in the algebra N is irreducible, of depth 2 and regular, giving a converse to the results of Enock and Nest. Finally we prove the equivalence of minimal and outer actions and we generalize a theorem of Yamanouchi: every integrable outer action with infinite fixed point algebra is a dual action.Comment: 37 pages, LaTeX 2

Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)

We survey Sorin Popa's recent work on Bernoulli actions. The paper was written on the occasion of the Bourbaki seminar. Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra, yielding the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II_1 factors with prescribed countable fundamental group.Comment: Minor correction

A new approach to induction and imprimitivity results

In the framework of locally compact quantum groups, we provide an induction procedure for unitary corepresentations as well as coactions on C*-algebras. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. We essentially use von Neumann algebraic techniques.Comment: We added an imprimitivity characterization of induced coactions. Minor typographical corrections as wel

One-cohomology and the uniqueness of the group measure space decomposition of a II_1 factor

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II_1 factor. We single out a large class of groups \Gamma, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of \Gamma the associated II_1 factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan-Peterson, but we replace the usage of Peterson's unbounded derivations by Thomas Sinclair's dilation into a one-parameter group of automorphisms.Comment: v2: minor changes, final version, to appear in Mathematische Annale