Subalgebras, subgroups, and singularity

This paper concerns the non-commutative analog of the Normal Subgroup Theorem for certain groups. Inspired by Kalantar-Panagopoulos, we show that all $\mathrm{\Gamma }$-invariant subalgebras of $L\mathrm{\Gamma }$ and $C_r^{\ast }(\mathrm{\Gamma })$ are ($\mathrm{\Gamma }$-) 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 $\mathrm{\Gamma }$-invariant intermediate von Neumann subalgebras $L^{\mathrm{\infty }}(X,\xi )\subset \mathcal{M}\subset L^{\mathrm{\infty }}(X,\xi )⋊\mathrm{\Gamma }$ in terms of the normal subgroups of $\mathrm{\Gamma }$.