Resumen de Fell bundles associated to groupoid morphisms

Valentin Deaconu, Alex Kumjian, Birant Ramazan

• 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 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.