Principal Actions on Topological Quivers and Associated Operator Dynamics

Abstract

We study topological quivers $Q$ admitting a free and proper action by a locally compact group $G$ together with their associated $C^*$-algebras. On the topological side, we provide a complete classification of topological quivers which admit such actions in terms of $G$-bundles over the vertex orbit space and an appropriate isomorphism of bundles over the edge orbits. Following the work by Deaconu, Kumjian, and Quigg on topological graphs, we construct an isomorphism between $C^*(Q/G)$ and Rieffel's fixed-point algebra $C^*(Q)^\alpha$, which is known to be Morita equivalent to $C^*(Q)\rtimes_rG$. Unlike the work with topological graphs, we use previously developed functoriality techniques to identify the isomorphism. We also examine many concrete examples of such group actions, including some exclusive to topological quivers, and the associated Morita equivalences.