The orbit spaces of groupoids whose $C^*$-algebras are CCR

TL;DR

This paper proves that for second countable locally compact Hausdorff groupoids with CCR C*-algebras, the orbits are necessarily closed, removing the previous requirement of amenability.

Contribution

It extends Clark's theorem by removing the amenability assumption, establishing that CCR C*-algebras imply closed orbits in this class of groupoids.

Findings

CCR C*-algebra implies closed orbits

Removed the amenability assumption from previous results

Applicable to second countable locally compact Hausdorff groupoids

Abstract

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