Simple subquotients of crossed products by abelian groups and twisted group algebras

TL;DR

This paper investigates conditions under which simple subquotients of crossed products by abelian groups are isomorphic to simple twisted group algebras, revealing their structure and relation to non-commutative tori.

Contribution

It provides new criteria for identifying simple subquotients of crossed products as twisted group algebras, extending Poguntke's theorem to a broader context.

Findings

Simple subquotients can be characterized as twisted group algebras under certain conditions.

Connected group $C^*$-algebras have simple subquotients that are either trivial or non-commutative tori.

The results unify and extend previous classifications of simple subquotients in group $C^*$-algebras.

Abstract

Motivated by work of Poguntke we study the question under what conditions simple subquotients of crossed products $A\rtimes_{\alpha}G$ by (twisted) actions of abelian groups $G$ are isomorphic to simple twisted group algebras of abelian groups. As a consequence, we recover a theorem of Poguntke's saying that the simple subquotients of group $C^*$-algebras of connected groups are either stably isomorphic to $\mathbb C$ or they are stably isomorphic to simple non-commutative tori.