Fell bundles over inverse semigroups and twisted etale groupoids

TL;DR

This paper constructs a twisted etale groupoid from a semi-abelian Fell bundle over an inverse semigroup, enabling classification of Cartan subalgebras in C*-algebras via groupoid methods.

Contribution

It introduces a method to recover a twisted etale groupoid from a semi-abelian Fell bundle, extending Renault's classification results.

Findings

Constructed a twisted etale groupoid from a semi-abelian Fell bundle.

Revealed a canonical way to recover the Fell bundle from the groupoid.

Extended Renault's classification of Cartan subalgebras using this framework.

Abstract

Given a saturated Fell bundle A over an inverse semigroup S which is semi-abelian in the sense that the fibers over the idempotents of S are commutative, we construct a twisted etale groupoid L such that A can be recovered from L in a canonical way. As an application we recover most of Renault's recent result on the classification of Cartan subalgebras of C*-algebras through twisted etale groupoids.