The primitive spectrum of C*-algebras of etale groupoids with abelian isotropy

TL;DR

This paper characterizes the primitive spectrum of C*-algebras associated with étale groupoids having abelian isotropy, providing a topological description that aids in understanding their ideal structures.

Contribution

It offers a topological description of the primitive spectrum for a class of C*-algebras from étale groupoids with abelian isotropy, confirming a conjecture by van Wyk and Williams.

Findings

Describes the primitive spectrum as a topological space for certain groupoid C*-algebras.

Provides a complete description of the ideal structure for these algebras.

Applies results to higher rank graph C*-algebras without sources.

Abstract

Given a Hausdorff locally compact \'etale groupoid $\mathcal G$, we describe as a topological space the part of the primitive spectrum of $C^*(\mathcal G)$ obtained by inducing one-dimensional representations of amenable isotropy groups of $\mathcal G$. When $\mathcal G$ is amenable, second countable, with abelian isotropy groups, our result gives the description of $\operatorname{Prim} C^*(\mathcal G)$ conjectured by van Wyk and Williams. This, in principle, completely determines the ideal structure of a large class of separable C$^*$-algebras, including the transformation group C$^*$-algebras defined by amenable actions of discrete groups with abelian stabilizers and the C$^*$-algebras of higher rank graphs. As an illustration we describe the primitive spectrum of the C$^*$-algebra of any row-finite higher rank graph without sources.