Group actions on topological graphs

Abstract

We define the action of a locally compact group $G$ on a topological graph $E$. This action induces a natural action of $G$ on the $C^{\ast }$-correspondence $\mathcal{H}(E)$ and on the graph $C^{\ast }$-algebra $C^{\ast }(E)$. If the action is free and proper, we prove that $C^{\ast }(E){⋊}_{r}G$ is strongly Morita equivalent to $C^{\ast }(E/G)$. We define the skew product of a locally compact group $G$ by a topological graph $E$ via a cocycle $c:E^1\to G$. The group acts freely and properly on this new topological graph $E{\times }_{c}G$. If $G$ is abelian, there is a dual action on $C^{\ast }(E)$ such that $C^{\ast }(E)⋊\hat{G}\cong C^{\ast }(E{\times }_{c}G)$. We also define the fundamental group and the universal covering of a topological graph.

We corrected a gap in the proof of Thm 5.6