W*-superrigidity of mixing Gaussian actions of rigid groups

Abstract

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.