The Primitive Ideal Space of Groupoid C*-Algebras for Groupoids with Abelian Isotropy

TL;DR

This paper investigates the topology of primitive ideal spaces in groupoid C*-algebras with abelian isotropy, extending known results and providing new insights for specific classes of groupoids.

Contribution

It offers a comprehensive analysis of primitive ideal spaces for groupoids with abelian isotropy, including cases with discontinuous isotropy maps and certain extensions.

Findings

Complete descriptions for groupoids with continuous isotropy maps.

Extension of known results to groupoids with jump discontinuities.

Analysis of unit space fixing extensions by abelian groups.

Abstract

We study the topology of the primitive ideal space of groupoid C*-algebras for groupoids with abelian isotropy. Our results include the known results for action groupoids with abelian stabilizers. Furthermore, we obtain complete results when the isotropy map is continuous except for jump discontinuities, and also when $G$ is a unit space fixing extension of a proper groupoid by an abelian group bundle. We hope that our methods will be a springboard to further results of this type.