Reduced C*-algebras of Fell bundles over inverse semigroups

TL;DR

This paper introduces a new way to construct reduced C*-algebras from Fell bundles over inverse semigroups, compares them with groupoid C*-algebras, and explores conditions for their equivalence.

Contribution

It defines the reduced C*-algebra for Fell bundles over inverse semigroups and analyzes when it matches the reduced groupoid C*-algebra.

Findings

Constructed a weak conditional expectation for Fell bundles.

Identified conditions under which reduced C*-algebras coincide.

Provided counterexamples to the equivalence.

Abstract

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.