The orbit space of groupoids whose $C^*$-algebras are GCR

TL;DR

This paper proves that for second countable locally compact Hausdorff groupoids with Haar systems, the GCR property of their $C^*$-algebras implies that the groupoid's orbits are locally closed, removing the need for amenability assumptions.

Contribution

It extends Clark's theorem by removing the amenability condition, establishing a direct link between GCR $C^*$-algebras and orbit closure properties in groupoids.

Findings

GCR $C^*$-algebras imply locally closed orbits

Removal of amenability assumption in existing theorems

Enhanced understanding of the structure of groupoid $C^*$-algebras

Abstract

Let $G$ be second countable locally compact Hausdorff groupoid with a continuous Haar system. We remove the assumption of amenability in a theorem by Clark about GCR groupoid $C^*$-algebras. We show that if the groupoid $C^*$-algebra of $G$ is GCR then the orbits of $G$ are locally closed.