Groupoid C*-algebras with Hausdorff Spectrum

TL;DR

This paper characterizes when the $C^*$-algebra of a second countable, locally compact Hausdorff groupoid with abelian stabilizers has a Hausdorff spectrum, linking it to stabilizer continuity and orbit space properties.

Contribution

It provides necessary and sufficient conditions for the spectrum of the groupoid $C^*$-algebra to be Hausdorff, focusing on stabilizer variation and orbit space topology.

Findings

Spectrum is Hausdorff iff stabilizers vary continuously in Fell topology.

Orbit space $G^{(0)}/G$ must be Hausdorff.

Convergence conditions in the dual stabilizer groupoid relate to spectrum Hausdorffness.

Abstract

Suppose $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabilizer subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid $C^*$-algebra to have Hausdorff spectrum. In particular we show that the spectrum of $C^*(G)$ is Hausdorff if and only if the stabilizers vary continuously with respect to the Fell topology, the orbit space $G^{(0)}/G$ is Hausdorff, and, given convergent sequences $\chi_i\to \chi$ and $\gamma_i\cdot\chi_i \to \omega$ in the dual stabilizer groupoid $\hat{S}$ where the $\gamma_i\in G$ act via conjugation, if $\chi$ and $\omega$ are elements of the same fiber then $\chi = \omega$