Properly proximal groups and their von Neumann algebras

TL;DR

This paper introduces properly proximal groups, a broad class including many important groups, and demonstrates their von Neumann algebras exhibit strong rigidity properties, with applications to actions of SL_d(Z).

Contribution

It defines properly proximal groups and proves their von Neumann algebras have unique Cartan subalgebras, leading to new rigidity results for certain group actions.

Findings

Properly proximal groups include non-amenable bi-exact groups and lattices in semi-simple Lie groups.

Crossed product II$_1$ factors from these groups have at most one weakly compact Cartan subalgebra.

First $W^*$-strong rigidity results for compact actions of $SL_d(b Z)$ for $d \,\geq\, 3$.

Abstract

We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.