Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles

TL;DR

This paper extends non-triviality conditions from group actions to Fell bundles over discrete groups, establishing criteria for simplicity, pure infiniteness, and aperiodicity of associated C*-algebras, with implications for their ideal structure and spectral properties.

Contribution

It introduces new equivalences and criteria for aperiodicity, topological freeness, and pure outerness in Fell bundles, generalizing known conditions from group actions to this broader setting.

Findings

Aperiodicity equals topological freeness under certain conditions.

C_r(B) is simple iff B is minimal and pointwise outer.

Aperiodicity is characterized by non-trivial Connes spectrum of fibres.

Abstract

We generalise various non-triviality conditions for group actions to Fell bundles over discrete groups and prove several implications between them. We also study sufficient criteria for the reduced section C*-algebra C_r(B) of a Fell bundle (B_g) to be strongly purely infinite. If the unit fibre A=B_e contains an essential ideal that is separable or of Type I, then the Fell bundle is aperiodic if and only if it is topologically free. If, in addition, G=Z or G=Z/p for a square-free number p, then these equivalent conditions are satisfied if and only if A detects ideals in C_r(B), if and only if A^+ \ {0} supports C_r(B) in the Cuntz sense. For G as above and arbitrary A, C_r(B) is simple if and only if the Fell bundle B is minimal and pointwise outer. In general, B is aperiodic if and only if each of its non-trivial fibres has a non-trivial Connes spectrum. If G is finite or if A contains an essential ideal that is of Type I or simple, then aperiodicity is equivalent to pointwise pure outerness.