Free actions of groups on separated graph C*-algebras

TL;DR

This paper investigates free group actions on separated graph C*-algebras, generalizing known results for ordinary graphs, and characterizes these actions via skew products and coactions, revealing structural properties of the associated C*-algebras.

Contribution

It extends the Gross-Tucker Theorem to separated graphs and describes the C*-algebras as crossed products, introducing new techniques involving Fell bundles and amalgamated free products.

Findings

Characterization of free actions via skew products.

C*-algebras as crossed products by coactions.

Existence of canonical Fell bundle structures.

Abstract

In this paper we study free actions of groups on separated graphs and their \cstar{}algebras, generalizing previous results involving ordinary (directed) graphs. We prove a version of the Gross-Tucker Theorem for separated graphs yielding a characterization of free actions on separated graphs via a skew product of the (orbit) separated graph by a group labeling function. Moreover, we describe the C*-algebras associated to these skew products as crossed products by certain coactions coming from the labeling function on the graph. Our results deal with both the full and the reduced C*-algebras of separated graphs. To prove our main results we use several techniques that involve certain canonical conditional expectations defined on the C*-algebras of separated graphs and their structure as amalgamated free products of ordinary graph C*-algebras. Moreover, we describe Fell bundles associated with the coactions of the appearing labeling functions. As a byproduct of our results, we deduce that the \cstar{}algebras of separated graphs always have a canonical Fell bundle structure over the free group on their edges.