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

Motivated by work of Poguntke we study the question under what conditions simple subquotients of crossed products $A{⋊}_{\alpha }G$ 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 $C^{\ast }$-algebras of connected groups are either stably isomorphic to $\mathbb{C}$ or they are stably isomorphic to simple non-commutative tori.