# Rigidity for von Neumann algebras given by locally compact groups and their crossed products

## TL;DR

This paper establishes the first rigidity and classification results for crossed product von Neumann algebras arising from actions of non-discrete, locally compact groups, focusing on simple Lie groups and their products.

## Contribution

It introduces novel rigidity theorems for von Neumann algebras associated with non-discrete group actions, extending classification to broader classes of locally compact groups.

## Key findings

- Unique Cartan subalgebra for actions of simple Lie groups
- W* strong rigidity for irreducible actions of product groups
- Results apply to nonamenable, weakly amenable groups in Ozawa's class S

## Abstract

We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S.