W*-superrigidity of mixing Gaussian actions of rigid groups

TL;DR

This paper extends W*-superrigidity results from Bernoulli actions to mixing Gaussian actions for certain rigid groups, showing that von Neumann algebra isomorphisms imply conjugacy of actions and group isomorphisms.

Contribution

It generalizes W*-superrigidity from Bernoulli to mixing Gaussian actions for w-rigid and non-amenable ICC product groups, establishing new rigidity phenomena.

Findings

Mixing Gaussian actions of w-rigid ICC groups are W*-superrigid.

Von Neumann algebra isomorphisms imply conjugacy of actions.

Results apply to non-amenable ICC product groups.

Abstract

We generalize W*-superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If \Gamma\ 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 \sigma\ of \Gamma\ is W*-superrigid. More precisely, if \rho\ is another free ergodic action of a group \Lambda\ such that the crossed-product von Neumann algebras associated with \rho\ and \sigma\ are isomorphic, then \Lambda\ and \Gamma\ are isomorphic, and the actions \rho\ and \sigma\ are conjugate. We prove a similar statement whenever \Gamma\ is a non-amenable ICC product of two infinite groups.