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

We investigate structural properties of the reduced cross-sectional algebra Cr⁎(B) of a Fell bundle BB over a discrete group G  . Conditions allowing one to determine the ideal structure of Cr⁎(B) are studied. Notions of aperiodicity, paradoxicality and BB-infiniteness for the Fell bundle BB are introduced and investigated by themselves and in relation to the partial dynamical system dual to BB. Several criteria of pure infiniteness of Cr⁎(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, Rørdam [47]; Giordano, Sierakowski [18] and Ortega, Pardo [39]. For exact, separable Fell bundles satisfying the residual intersection property primitive ideal space of Cr⁎(B) is determined. The results of the paper are shown to be optimal when applied to graph C⁎C⁎-algebras. Applications to a class of Exel–Larsen crossed products are presented.