Inverse semigroup actions on groupoids

Abstract

We de fine 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 nonHausdorff 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 stabilization trick does not generalize to non-Hausdorff groupoids.