## Free actions of groups on separated graph $C^*$-algebras

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- by Pere Ara, Alcides Buss and Ado Dalla Costa PDF
- Trans. Amer. Math. Soc.
**376**(2023), 2875-2919 Request permission

## 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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## Additional Information

**Pere Ara**- Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Barcelona, Spain; and Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès, Barcelona, Spain
- MR Author ID: 206418
- ORCID: 0000-0003-3739-9599
- Email: para@mat.uab.cat
**Alcides Buss**- Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis-SC, Brazil
- MR Author ID: 827256
- ORCID: 0000-0001-6796-9818
- Email: alcides.buss@ufsc.br
**Ado Dalla Costa**- Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis-SC, Brazil
- ORCID: 0000-0001-8029-2995
- Email: adodallacosta@hotmail.com
- Received by editor(s): April 19, 2022
- Received by editor(s) in revised form: September 12, 2022
- Published electronically: January 23, 2023
- Additional Notes: This work has been supported by CNPq/Humboldt-CAPES–Brazil. The first named author was partially supported by DGI-MINECO-FEDER grant PID2020-113047GB-I00, and the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R$\&$D (CEX2020-001084-M)
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 2875-2919 - MSC (2020): Primary 46L55, 22D35
- DOI: https://doi.org/10.1090/tran/8839
- MathSciNet review: 4557884