Fell bundles associated to groupoid morphisms

Abstract

Given a continuous open surjective morphism π : G → H of étale groupoids with amenable kernel, we construct a Fell bundle E over H and prove that its C* -algebra Cr* (E) is isomorphic to Cr*(G). This is related to results of Fell concerning C*-algebraic bundles over groups. The case H = X, a locally compact space, was treated earlier by Ramazan. We conclude that C* (G) is strongly Morita equivalent to a crossed product, the C*-algebra of a Fell bundle arising from an action of the groupoid H on a C*-bundle over H0 We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. We also prove a structure theorem for abelian Fell bundles.