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

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.