Rigidity for von Neumann algebras

TL;DR

This paper surveys recent advances in the classification of von Neumann algebras, focusing on superrigidity, unique Cartan subalgebras, and applications to orbit equivalence, highlighting key rigidity results for group-related algebras.

Contribution

It provides a comprehensive overview of new rigidity results, including superrigidity theorems and classification techniques for von Neumann algebras from group actions.

Findings

Identification of classes of superrigid actions

Existence of unique Cartan subalgebras in certain II$_1$ factors

Applications of cocycle superrigidity to orbit equivalence

Abstract

We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W$^*$-superrigid) actions that can be completely recovered from their von Neumann algebras and II$_1$ factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.