On the amenable subalgebras of group von Neumann algebras

TL;DR

This paper investigates the structure of sub-von Neumann algebras within group von Neumann algebras, revealing the existence of a maximal invariant amenable subalgebra and introducing invariant probability measures to analyze their properties.

Contribution

It introduces the concept of invariant probability measures on sub-algebras and proves the existence of a maximal invariant amenable subalgebra in group von Neumann algebras.

Findings

Existence of a maximal invariant amenable subalgebra in $L\Gamma$.

Invariant probability measures are supported on the maximal amenable invariant subalgebra.

Amenable IRAs are analogous to invariant random subgroups.

Abstract

We approach the study of sub-von Neumann algebras of the group von Neumann algebra $L\Gamma$ for countable groups $\Gamma$ from a dynamical perspective. It is shown that $L(\Gamma)$ admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of sub-algebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant sub-algebra.