Twisted actions and regular Fell bundles over inverse semigroups

TL;DR

This paper introduces a new concept of twisted actions of inverse semigroups, establishing a bijective correspondence with regular Fell bundles, and applies this to describe twisted etale groupoid C*-algebras in terms of crossed products.

Contribution

It generalizes previous twisted action definitions, providing a structure classification of regular Fell bundles over inverse semigroups and connecting them to twisted etale groupoid C*-algebras.

Findings

Bijective correspondence between twisted actions and regular Fell bundles

Generalization of Sieben's twisted actions to a broader context

Description of twisted etale groupoid C*-algebras via crossed products

Abstract

We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain regular Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the stable Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby-Smith twisted actions in the group case. As an application we describe twisted etale groupoid C*-algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sieben's twisted actions essentially correspond to twisted etale groupoids with topologically trivial twists.