Inverse semigroup actions on 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 étale 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 étale 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.

59 pages. Proofreading with minor changes. Version accepted for publication at Rocky Mountain Journal

References in corpus (5)

• Renault's Equivalence Theorem for Groupoid Crossed Products
• Noncommutative Cartan sub-algebras of C*-algebras
• Groupoids in categories with pretopology
• Tannaka duality for proper Lie groupoids
• Topological construction of $C^{\ast }$-correspondences for groupoid $C^{\ast }$-algebras

Cited by in corpus (16)

• Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness
• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations
• Reduced C*-algebras of Fell bundles over inverse semigroups
• Stone duality and quasi-orbit spaces for generalised C*-inclusions
• Iterated crossed products for groupoid fibrations
• A universal property for groupoid C*-algebras. I
• Quantales and Fell bundles
• Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids
• Linear structures on locales
• Representations of *-algebras by unbounded operators: C*-hulls, local-global principle, and induction
• Strongly graded groupoid and directed graph algebras
• Inverse Semigroupoid Actions and Representations
• On Fell bundles over inverse semigroups and their left regular representations
• The bicategory of groupoid correspondences
• Inclusions of C*-algebras of graded groupoids
• Localised Hilbert modules and weak noncommutative Cartan pairs