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∗-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 C∗-algebras of separated graphs always have a canonical Fell bundle structure over the free group on their edges. 2023 American Mathematical Society.