Pure infiniteness and ideal structure of C*-algebras associated to Fell bundles

We investigate structural properties of the reduced cross-sectional algebra C-r*(B) of a Fell bundle B over a discrete group G. Conditions allowing one to determine the ideal structure of C-r*(B) are studied. Notions of aperiodicity, paradoxicality and B -infiniteness for the Fell bundle B are introduced and investigated by themselves and in relation to the partial dynamical system dual to B. Several criteria of pure infiniteness of C-r*(B) are given. It is shown that they generalize and unify corresponding results obtained in the context of crossed products, by the following duos: Laca, Spielberg [34]; Jolissaint, Robertson [21]; Sierakowski, Rordam [47]; Giordano, Sierakowski [18] and Ortega, Pardo [39]. For exact, separable Fell bundles satisfying the residual intersection property primitive ideal space of C-r*(B) is determined. The results of the paper are shown to be optimal when applied to graph C*-algebras. Applications to a class of Exel Larsen crossed products are presented. (C) 2016 Elsevier Inc. All rights reserved.