This paper concerns the non-commutative analog of the Normal Subgroup Theorem for certain groups. Inspired by Kalantar-Panagopoulos, we show that all $\Gamma$-invariant subalgebras of $L\Gamma$ and $C^*_r(\Gamma)$ are ($\Gamma$-) 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 $\Gamma$-invariant intermediate von Neumann subalgebras $L^{ \infty}(X,\xi)\subset\mathcal{ M}\subset L^{ \infty}(X,\xi)\rtimes\Gamma$ in terms of the normal subgroups of $\Gamma$.
Categories: math.OA math.DS math.GR