Free actions of groups on separated graph C^-algebras

• Published in: Transactions of the American Mathematical Society 2023-04, Vol.376 (4), p.2875
• Main Authors: Pere Ara, Alcides Buss, Ado Dalla Costa
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/8839

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.