Fell bundles associated to groupoid morphisms - Deaconu, Valentin; Kumjian, Alex; Ramazan, Birant (MATHEMATICA SCANDINAVICA, 2008)
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 C^*_r(E) is isomorphic to C^*_r(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^*_r(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 H^0. 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.