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Transactions of the American Mathematical Society Series B

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$\mathrm {C^*}$-algebras, groupoids and covers of shift spaces


Authors: Kevin Aguyar Brix and Toke Meier Carlsen
Journal: Trans. Amer. Math. Soc. Ser. B 7 (2020), 134-185
MSC (2020): Primary 46L55, 37A55, 37B10
DOI: https://doi.org/10.1090/btran/53
Published electronically: October 30, 2020
MathSciNet review: 4168660
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Abstract:

To every one-sided shift space $\mathsf {X}$ we associate a cover $\widetilde {\mathsf {X}}$, a groupoid $\mathcal {G}_\mathsf {X}$ and a $\mathrm {C^*}$-algebra $\mathcal {O}_\mathsf {X}$. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between $\mathsf {X}$ and $\mathsf {Y}$ in terms of isomorphism of $\mathcal {G}_\mathsf {X}$ and $\mathcal {G}_\mathsf {Y}$, and diagonal-preserving $^*$-isomorphism of $\mathcal {O}_\mathsf {X}$ and $\mathcal {O}_\mathsf {Y}$. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces $\Lambda _\mathsf {X}$ and $\Lambda _\mathsf {Y}$ in terms of isomorphism of the stabilized groupoids $\mathcal {G}_\mathsf {X}\times \mathcal {R}$ and $\mathcal {G}_\mathsf {Y}\times \mathcal {R}$, and diagonal-preserving $^*$-isomorphism of the stabilized $\mathrm {C^*}$-algebras $\mathcal {O}_\mathsf {X}\otimes \mathbb {K}$ and $\mathcal {O}_\mathsf {Y}\otimes \mathbb {K}$. Our strategy is to lift relations on the shift spaces to similar relations on the covers.

Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of $\mathsf {X}$ from the pair $(\mathcal {O}_\mathsf {X}, C(\mathsf {X}))$, and the flow equivalence class of $\Lambda _\mathsf {X}$ from the pair $(\mathcal {O}_\mathsf {X}\otimes \mathbb {K}, C(\mathsf {X})\otimes c_0)$. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.


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Additional Information

Kevin Aguyar Brix
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia
MR Author ID: 1338809
ORCID: 0000-0003-1237-6446
Email: kabrix.math@fastmail.com

Toke Meier Carlsen
Affiliation: Department of Science and Technology, University of the Faroe Islands, Vestara Bryggja 15, FO-100 Tórshavn, the Faroe Islands
MR Author ID: 685180
ORCID: 0000-0002-7981-7130
Email: toke.carlsen@gmail.com

Keywords: Symbolic dynamics, shift spaces, groupoids, $\mathrm C^*$-algebras
Received by editor(s): October 8, 2019
Received by editor(s) in revised form: March 9, 2020
Published electronically: October 30, 2020
Additional Notes: The first named author was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) and the Carlsberg Foundation through an Internationalisation Fellowship.
The second named author was supported by Research Council Faroe Islands through the project “Using graph $\mathrm {C^*}$-algebras to classify graph groupoids”.
Article copyright: © Copyright 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)