On the amenable subalgebras of group von Neumann algebras

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.