Exactness and Fell bundles with the approximation property over inverse semigroups

TL;DR

This paper proves that the exactness of the reduced cross-sectional algebra of a Fell bundle over an inverse semigroup depends precisely on the exactness of its unit fiber, generalizing previous groupoid results.

Contribution

It establishes a necessary and sufficient condition for the exactness of Fell bundle cross-sectional algebras over inverse semigroups, extending known groupoid theorems.

Findings

Reduced cross-sectional algebra is exact iff the unit fiber is exact

Generalizes a recent groupoid action result

Reproves some results on Fell bundle ideals

Abstract

We prove that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. This generalizes a recent result of the first-named author for actions of second countable locally compact Hausdorff groupoids on separable $C^*$-algebras. Along the way, we reprove some results of Kwa\'sniewski--Meyer on Fell bundle ideals.