Subalgebras, subgroups, and singularity

TL;DR

This paper extends the Normal Subgroup Theorem to non-commutative groups, showing all invariant subalgebras are co-amenable and describing intermediate von Neumann subalgebras via normal subgroups.

Contribution

It introduces a singularity framework to characterize invariant subalgebras and provides a non-commutative analog of the classical theorem for specific groups.

Findings

All invariant subalgebras are co-amenable.

Intermediate von Neumann subalgebras correspond to normal subgroups.

Framework applies to groups with a singularity phenomenon.

Abstract

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 $\Gamma$.