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

Abstract

Given a Hausdorff locally compact étale groupoid $\mathcal{G}$, we describe as a topological space the part of the primitive spectrum of $C^{\ast }(\mathcal{G})$ 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 $\mathrm{Prim}{C}^{\ast }(\mathcal{G})$ conjectured by van Wyk and Williams. This, in principle, completely determines the ideal structure of a large class of separable C${}^{\ast }$-algebras, including the transformation group C${}^{\ast }$-algebras defined by amenable actions of discrete groups with abelian stabilizers and the C${}^{\ast }$-algebras of higher rank graphs. As an illustration we describe the primitive spectrum of the C${}^{\ast }$-algebra of any row-finite higher rank graph without sources.

34 pages; v3: the revised version contains various improvements, including that it no longer depends on the main results of arXiv:2308.10768 and gives a description of the primitive spectrum of C*-algebras of higher rank graphs in terms of maximal tails, periodicity groups and finite paths; v2: minor changes and corrections