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

TL;DR

This paper characterizes the primitive ideal space of C*-algebras from amenable étale groupoids with isotropy groups of local polynomial growth, providing explicit descriptions and computations for specific examples.

Contribution

It offers a detailed topological and representation-theoretic description of primitive ideals for a broad class of groupoid C*-algebras, including special cases with simplified structures.

Findings

Primitive spectrum described in terms of groupoid topology and isotropy representations

Simplifications occur for FC-hypercentral isotropy groups or certain transformation groupoids

Explicit computation of primitive spectrum for a specific SL_3(Z) action

Abstract

Given an amenable second countable Hausdorff locally compact \'etale groupoid $\mathcal G$ such that each isotropy group $\mathcal G^x_x$ has local polynomial growth, we give a description of $\operatorname{Prim} C^*(\mathcal G)$ 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 $\Gamma\ltimes X$ defined by an action $\Gamma\curvearrowright X$ with locally finite stabilizers. To illustrate the class of C$^*$-algebras for which our results can provide a complete description of the ideal structure, we compute the primitive spectrum of $\mathrm{SL}_3(\mathbb Z)\ltimes C_0(\mathrm{SL}_3(\mathbb R)/U_3(\mathbb R))$, where $U_3(\mathbb R)$ is the group of unipotent upper triangular matrices.