Invariant random subgroups and action versus representation maximality

Abstract

We show that weak containment of free ergodic measure-preserving actions of ${\mathbf{F}}_{\infty }$ is not equivalent to weak containment of the corresponding Koopman representations. This result is based on the construction of an invariant random subgroup of ${\mathbf{F}}_{\infty }$ which is supported on the maximal actions.