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 $\mathrm C^*$-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.
