Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles
Volume 241 / 2018
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 -algebra \mathrm C^*_\mathrm r(\mathcal{B}) of a Fell bundle \mathcal{B}=(B_g)_{g\in 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 \mathcal{B} is aperiodic if and only if \mathcal{B} is topologically free. If, in addition, G=\mathbb Z or G=\mathbb Z/p for a square-free number p, then these equivalent conditions are satisfied if and only if A detects ideals in \mathrm C^*_\mathrm r(\mathcal{B}), if and only if A^+\setminus\{0\} supports \mathrm C^*_\mathrm r(\mathcal{B})^+\setminus\{0\} in the Cuntz sense. For G as above and for arbitrary A, \mathrm C^*_\mathrm r(\mathcal{B}) is simple if and only if \mathcal{B} is minimal and pointwise outer. In general, \mathcal{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.