The ideal structure of C*-algebras of etale groupoids with isotropy groups of local polynomial growth

Abstract

Given an amenable second countable Hausdorff locally compact étale groupoid $\mathcal{G}$ such that each isotropy group ${\mathcal{G}}_x^x$ has local polynomial growth, we give a description of $\mathrm{Prim}{C}^{\ast }(\mathcal{G})$ as a topological space in terms of the topology on $\mathcal{G}$ and representation theory of the isotropy groups and their subgroups. The description simplifies when either the isotropy groups are FC-hypercentral or $\mathcal{G}$ is the transformation groupoid $\Gamma ⋉X$ defined by an action $\Gamma \curvearrowright X$ with locally finite stabilizers. To illustrate the class of C${}^{\ast }$-algebras for which our results can provide a complete description of the ideal structure, we compute the primitive spectrum of ${\mathrm{SL}}_3(\mathbb{Z})⋉C_0({\mathrm{SL}}_3(\mathbb{R})/U_3(\mathbb{R}))$, where $U_3(\mathbb{R})$ is the group of unipotent upper triangular matrices.

46 pages; v4: minor corrections and changes in exposition, more references; v3: a discussion of essentially principal groupoids and the ideal intersection property added; v2: minor corrections and improvements, a discussion of type I case added