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

Abstract

We investigate structural properties of the reduced cross-sectional algebra $C^*_r(\mathcal{B})$ of a Fell bundle $\mathcal{B}$ over a discrete group $G$. Conditions allowing one to determine the ideal structure of $C^*_r(\mathcal{B})$ are studied. Notions of aperiodicity, paradoxicality and $\mathcal{B}$-infiniteness for the Fell bundle $\mathcal{B}$ are introduced and investigated by themselves and in relation to the partial dynamical system dual to $\mathcal{B}$. Several criteria of pure infiniteness of $C^*_r(\mathcal{B})$ 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; Jolissaint, Robertson; Sierakowski, Rørdam; Giordano, Sierakowski and Ortega, Pardo. For exact, separable Fell bundles satisfying the residual intersection property primitive ideal space of $C^*_r(\mathcal{B})$ 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.