Pure infiniteness and ideal structure of $C^{\ast }$-algebras associated to Fell bundles

Abstract

We investigate structural properties of the reduced cross-sectional algebra $C_r^{\ast }(\mathcal{B})$ of a Fell bundle $\mathcal{B}$ over a discrete group $G$. Conditions allowing one to determine the ideal structure of $C_r^{\ast }(\mathcal{B})$ 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 $C_r^{\ast }(\mathcal{B})$ 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 $C_r^{\ast }(\mathcal{B})$ is determined. The results of the paper are shown to be optimal when applied to graph $C^{\ast }$-algebras. Applications to a class of Exel-Larsen crossed products are presented.

This is the version accepted to Journal of Mathematical Analysis and Applications. Pure infiniteness criteria have been generalized. In particular, now they unify the corresponding results of Jolissaint, Robertson and Sierakowski, Rørdam

References in corpus (4)

• The ideal structure of reduced crossed products
• Crossed products by endomorphisms of $C_0(X)$-algebras
• Crossed product of a C*-algebra by an endomorphism, coefficient algebras and transfer operators
• Ideal structure of crossed products by endomorphisms via reversible extensions of $C^{\ast }$-dynamical systems

Cited by in corpus (6)

• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• Nica-Toeplitz algebras associated with product systems over right LCM semigroups
• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations
• Stone duality and quasi-orbit spaces for generalised C*-inclusions
• Crossed products by endomorphisms of $C_0(X)$-algebras
• Positive definiteness and Fell bundles over discrete groups