Inverse semigroup actions on groupoids

TL;DR

This paper introduces a framework for inverse semigroup actions on topological groupoids via partial equivalences, leading to new constructions of Fell bundles and insights into their properties, especially in non-Hausdorff contexts.

Contribution

It defines inverse semigroup actions on topological groupoids using partial equivalences and constructs associated Fell bundles, extending the theory to non-Hausdorff groupoids.

Findings

Constructed saturated Fell bundles over non-Hausdorff étale groupoids.

Showed that these Fell bundles are generally not Morita equivalent to automorphism actions.

Demonstrated that the Packer-Raeburn Stabilisation Trick does not extend to non-Hausdorff groupoids.

Abstract

We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff \'etale groupoids. We interpret these as actions on C*-algebras by Hilbert bimodules and describe the section algebras of these Fell bundles. Our constructions give saturated Fell bundles over non-Hausdorff \'etale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms. That is, the Packer-Raeburn Stabilisation Trick does not generalise to non-Hausdorff groupoids.