Group actions on topological graphs

TL;DR

This paper explores how locally compact groups act on topological graphs and their associated $C^*$-algebras, establishing Morita equivalences and dualities, and introducing concepts like fundamental groups and coverings for topological graphs.

Contribution

It introduces a framework for group actions on topological graphs and their $C^*$-algebras, including Morita equivalence results and duality for abelian groups.

Findings

Morita equivalence between crossed products and quotient graph $C^*$-algebras

Duality between crossed products and skew product graphs for abelian groups

Definitions of fundamental group and universal covering for 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 $C^*$-correspondence ${\mathcal H}(E)$ and on the graph $C^*$-algebra $C^*(E)$. If the action is free and proper, we prove that $C^*(E)\rtimes_r G$ is strongly Morita equivalent to $C^*(E/G)$. 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 $E\times_cG$. If $G$ is abelian, there is a dual action on $C^*(E)$ such that $C^*(E)\rtimes \hat{G}\cong C^*(E\times_cG)$. We also define the fundamental group and the universal covering of a topological graph.