Subalgebras, subgroups, and singularity

Abstract

This paper concerns the non-commutative analog of the Normal Subgroup Theorem for certain groups. Inspired by Kalantar-Panagopoulos, we show that all $Γ$-invariant subalgebras of $LΓ$ and $C^*_r(Γ)$ are ($Γ$-) co-amenable. The groups we work with satisfy a singularity phenomenon described in Bader-Boutonnet-Houdayer-Peterson. The setup of singularity allows us to obtain a description of $Γ$-invariant intermediate von Neumann subalgebras $L^{\infty}(X,ξ)\subset\mathcal{M}\subset L^{\infty}(X,ξ)\rtimesΓ$ in terms of the normal subgroups of $Γ$.