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

TL;DR

The paper develops a framework to analyze simple subquotients of crossed products by abelian groups, showing they are Morita equivalent to twisted group algebras of abelian groups via a decomposition by subquotients. Building on the Mackey–Rieffel–Green machinery, it reduces the problem to twisted group algebras $C^*(H, au)$ and, for abelian $G$, to continuous-field structures over symmetry groups $S_ au$, with fibers that are simple twisted group algebras on $G/S_ au$; under compactly generated, the fibers align with non-commutative tori or elementary algebras. The main results yield Poguntke-type dichotomies for simple quotients of group $C^*$-algebras of connected (and almost connected) groups, showing that simple subquotients are either compact operators or Morita equivalent to simple non-commutative tori of dimension $ ext{≥}2$, and extend these insights to broad classes of crossed products via smoothness and type I hypotheses. The work clarifies the spectral and Morita-theoretic structure of simple subquotients, with implications for understanding the primitive ideal spaces and their Hausdorff properties in twisted settings.

Abstract

Motivated by work of Poguntke we study the question under what conditions simple subquotients of crossed products $A\rtimes_αG$ 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.

Theorems & Definitions (75)

• Theorem 1.1: Poguntke
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Example 2.5
• Example 2.6
• Remark 2.10
• Theorem 2.11: Ech:induced*Theorem