Free actions of groups on separated graph 𝐶*-algebras

Ara, Pere; Buss, Alcides; Costa, Ado

doi:10.1090/tran/8839

Free actions of groups on separated graph $C^*$-algebras
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by Pere Ara, Alcides Buss and Ado Dalla Costa;

Trans. Amer. Math. Soc. 376 (2023), 2875-2919

DOI: https://doi.org/10.1090/tran/8839

Published electronically: January 23, 2023

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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.

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