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Given a saturated Fell bundle A over an inverse semigroup S which issemi-abelian in the sense that the fibers over the idempotents of S arecommutative, we construct a twisted etale groupoid L such that A can berecovered from L in a canonical way. As an application we recover most ofRenault's recent result on the classification of Cartan subalgebras ofC*-algebras through twisted etale groupoids.

Fell bundles over inverse semigroups and twisted etale groupoids