On Exel-Pardo algebras

TL;DR

This paper extends the construction of Exel-Pardo algebras from discrete groups acting on finite graphs to locally compact groups acting on topological graphs, introducing a new class of $C^*$-algebras via cocycles.

Contribution

It generalizes the Exel-Pardo algebra construction to a broader setting involving locally compact groups and topological graphs, using cocycles to define new $C^*$-correspondences.

Findings

Constructed $C^*$-correspondences for locally compact group actions.

Established the associated Cuntz-Pimsner algebras as generalizations.

Provided a framework for analyzing group actions on topological graphs.

Abstract

We generalize a recent construction of Exel and Pardo, from discrete groups acting on finite directed graphs to locally compact groups acting on topological graphs. To each cocycle for such an action, we construct a $C^*$-correspondence whose associated Cuntz-Pimsner algebra is the analog of the Exel-Pardo $C^*$-algebra.