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 $L\left(\Gamma \right)$ for countable groups Γ from a dynamical perspective. It is shown that $L\left(\Gamma \right)$ admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of subalgebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant subalgebra.