Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T11:32:47.281Z Has data issue: false hasContentIssue false
  • English
  • Français

On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz–Krieger algebras

Part of: Selfadjoint operator algebras Topological dynamics Ergodic theory

Published online by Cambridge University Press:  17 May 2021

KENGO MATSUMOTO
KENGO MATSUMOTO*
Affiliation:
Department of Mathematics, Joetsu University of Education, Joetsu943-8512, Japan
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials.

Keywords

Cuntz–Krieger algebragauge actiontopological conjugacytopological Markov shift

MSC classification

Primary: 37A55: Relations with the theory of $C^*$-algebras 46L55: Noncommutative dynamical systems
Secondary: 46L35: Classifications of $C^*$-algebras 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2575 - 2582
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brix, K. A. and Carlsen, T. M.. Cuntz–Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids. J. Aust. Math. Soc. 109 (2020), 289–298.CrossRefGoogle Scholar
Brownlowe, N., Carlsen, T. M. and Whittaker, M. F.. Graph algebras and orbit equivalence. Ergod. Th. & Dynam. Sys. 37(2017), 389–417.CrossRefGoogle Scholar
Carlsen, T. M., Eilers, S., Ortega, E. and Restorff, G.. Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids. J. Math. Anal. Appl. 469(2019), 1088–1110.CrossRefGoogle Scholar
Carlsen, T. M. and Rout, J.. Diagonal-preserving gauge invariant isomorphisms of graph ${C}^{\ast }$ -algebras. J. Funct. Anal. 273(2017), 2981–2993.CrossRefGoogle Scholar
Carlsen, T. M., Ruiz, E. and Sims, A.. Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph ${C}^{\ast }$ -algebras and Leavitt path algebras. Proc. Amer. Math. Soc. 145(2017), 1581–1592.CrossRefGoogle Scholar
Cuntz, J. and Krieger, W.. A class of ${C}^{\ast }$ -algebras and topological Markov chains. Invent. Math. 56(1980), 251–268.CrossRefGoogle Scholar
Kitchens, B. P.. Symbolic Dynamics. Springer, Berlin, 1998.CrossRefGoogle Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Matsumoto, K.. On automorphisms of ${C}^{\ast }$ -algebras associated with subshifts. J. Operator Theory 44(2000), 91–112.Google Scholar
Matsumoto, K.. Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras. Pacific J. Math. 246(2010), 199–225.CrossRefGoogle Scholar
Matsumoto, K.. Strongly continuous orbit equivalence of one-sided topological Markov shifts. J. Operator Theory 74(2015), 101–127.CrossRefGoogle Scholar
Matsumoto, K.. Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras. Proc. Amer. Math. Soc. 145(2017), 1131–1140.CrossRefGoogle Scholar
Matsumoto, K.. Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras. Math. Z. 285(2017), 121–141.CrossRefGoogle Scholar
Matsumoto, K.. Relative Morita equivalence of Cuntz–Krieger algebras and flow equivalence of topological Markov shifts. Trans. Amer. Math. Soc. 370(2018), 7011–7050.CrossRefGoogle Scholar
Matsumoto, K.. State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras. Dyn. Syst. 34(2019), 93–112.CrossRefGoogle Scholar
Matsumoto, K.. Simple purely infinite ${C}^{\ast }$ -algebras associated with normal subshifts. Preprint, 2020, arXiv:2003.11711v2 [mathOA].Google Scholar
Matsumoto, K.. One-sided topological conjugacy of normal subshifts and gauge actions on the associated ${C}^{\ast }$ -algebras. Preprint, 2021, arXiv:2104.02203v2 [mathOA].CrossRefGoogle Scholar
Matsumoto, K. and Matui, H.. Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras. Kyoto J. Math. 54(2014), 863–878.CrossRefGoogle Scholar
Matsumoto, K. and Matui, H.. Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions. Ergod. Th. & Dynam. Sys. 36(2016), 1557–1581.CrossRefGoogle Scholar