Free actions of groups on separated graph 𝐶*-algebras

Abstract

In this paper we study free actions of groups on separated graphs and their C ∗ C^* -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 ∗ 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 ∗ C^* -algebras of separated graphs. To prove our main results we use several techniques that involve certain canonical conditional expectations defined on the C ∗ C^* -algebras of separated graphs and their structure as amalgamated free products of ordinary graph C ∗ 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 C ∗ C^* -algebras of separated graphs always have a canonical Fell bundle structure over the free group on their edges.