Classification and rigidity for von Neumann algebras

Abstract

We survey some recent progress on the classication of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. In particular, we present results which provide classes of (W-superrigid) groups and actions that can be entirely reconstructed from their von Neumann algebras. We also discuss the recent finding of several large families of II${}_1$ factors that have a unique group measure space decomposition.