Exactness and Fell bundles with the approximation property over inverse semigroups

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 $C^{\ast }$-algebras. Along the way, we reprove some results of Kwaśniewski--Meyer on Fell bundle ideals.

arXiv comment18 pages. Minor changes. Adjusted the title to avoid terminological conflicts with Kwaśniewski--Meyer's definition of exact Fell bundles