Journal of Operator Theory

Volume 67, Issue 1, Winter 2012 pp. 153-205.

Fell bundles over inverse semigroups and twisted etale groupoids

Authors: Alcides Buss (1) and Ruy Exel (2)
Author institution: (1) Departamento de Matematica, Universidade Federal de Santa Catarina, 88.040-900 Florianopolis-SC, Brasil
(2) Departamento de Matematica, Universidade Federal de Santa Catarina, 88.040-900 Florianopolis-SC, Brasil

Summary: 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 $(\G,\Sigma)$ such that $\A$ can be recovered from $(\G,\Sigma)$ 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.

Keywords: Fell bundle, inverse semigroup, twisted etale groupoid, upper-semicontinuous Banach bundle, crossed product, Cartan subalgebra.

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