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Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras

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We will study circle actions on Cuntz–Krieger algebras trivially acting on its canonical maximal abelian \(C^*\)-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.

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References

  1. Boyle, M., Handelman, D.: Orbit equivalence, flow equivalence and ordered cohomology. Isr. J. Math. 95, 169–210 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bowen, R., Franks, J.: Homology for zero-dimensional nonwandering sets. Ann. Math. 106, 73–92 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brown, L.G.: Stable isomorphism of hereditary subalgebras of \(C^*\)-algebras. Pac. J. Math. 71, 335–348 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brown, L.G., Green, P., Rieffel, M.A.: Stable isomorphism and strong Morita equivalence of \(C^*\)-algebras. Pac. J. Math. 71, 349–363 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Combes, F.: Crossed products and Morita equivalence. Proc. Lond. Math. Soc. 49, 289–306 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crocker, D., Kumjian, A., Raeburn, I., Williams, D.P.: An equivariant Brauer group and actions of groups of \(C^*\)-algebras. J. Funct. Anal. 146, 151–184 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cuntz, J.: Automorphisms of certain simple \(C^*\)-algebras. In: Streit, L. (ed.) Quantum Fields-Algebras, Processes. Springer, Wien-New York (1980)

  8. Cuntz, J., Krieger, W.: A class of \(C^*\)-algebras and topological Markov chains. Invent. Math. 56, 251–268 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Franks, J.: Flow equivalence of subshifts of finite type. Ergod. Theory Dyn. Syst. 4, 53–66 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kitchens, B.P.: Symbolic Dynamics. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  11. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  12. Matsumoto, K.: On automorphisms of \(C^*\)-algebras associated with subshifts. J. Oper. Theory 44, 91–112 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Matsumoto, K.: Strong shift equivalence of symbolic dynamical systems and Morita equivalence of \(C^*\)-algebras. Ergod. Theory Dyn. Syst. 24, 199–215 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Matsumoto, K.: Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras. Pac. J. Math. 246, 199–225 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Matsumoto, K.: Classification of Cuntz–Krieger algebras by orbit equivalence of topological Markov shifts. Proc. Am. Math. Soc. 141, 2329–2342 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Matsumoto, K.: Strongly continuous orbit equivalence of one-sided topological Markov shifts. J. Oper. Theory 74, 101–127 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Matsumoto, K., Matui, H.: Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras. Kyoto J. Math. 54, 863–878 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Matsumoto, K., Matui, H.: Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions. Preprint, arXiv:1403.0719, to appear in Ergod. Theory Dyn. Syst

  19. Matui, H.: Homology and topological full groups of étale groupoids on totally disconnected spaces. Proc. Lond. Math. Soc. 104, 27–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matui, H.: Topological full groups of one-sided shifts of finite type. J. Reine Angew. Math. 705, 35–84 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Muhly, P.S., Pask, D., Tomforde, M.: Strong shift equivalence of \(C^*\)-correspondences. Isr. J. Math. 167, 315–346 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Parry, W., Sullivan, D.: A topological invariant for flows on one-dimensional spaces. Topology 14, 297–299 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Poon, Y.T.: A K-theoretic invariant for dynamical systems. Trans. Am. Math. Soc. 311, 513–533 (1989)

    MathSciNet  Google Scholar 

  24. Raeburn, I., Williams, D.P.: Morita equivalence and continuous-trace \(C^*\)-algebras. Mathematical surveys and monographs, vol. 60. American Mathematical Society, Providence (1998)

  25. Renault, J.: A Groupoid Approach to \(C^*\)-Algebras, Lecture Notes in Math, vol. 793. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

  26. Rieffel, M.A.: Induced representations of \(C^*\)-algebras. Adv. Math. 13, 176–257 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rieffel, M.A.: Morita equivalence for \(C^*\)-algebras and \(W^*\)-algebras. J. Pure Appl. Algebra 5, 51–96 (1974)

    Article  MATH  Google Scholar 

  28. Rørdam, M.: Classification of Cuntz–Krieger algebras. K-theory 9, 31–58 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tomforde, M.: The Graph Algebra Problem Page: List of Open Problems. http://www.math.uh.edu/tomforde/GraphAlgebraProblems/ListOfProblems.html

  30. Williams, R.F.: Classification of subshifts of finite type. Ann. Math. 98, pp. 120-153 (1973). Erratum: Ann. Math. 99, pp. 380-381 (1974)

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Acknowledgments

The author would like to thank the referee for his careful reading the manuscript and various suggestions. This work was supported by JSPS KAKENHI Grant Numbers 23540237 and 15K04896.

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  1. Department of Mathematics, Joetsu University of Education, Joetsu, 943-8512, Japan

    Kengo Matsumoto

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  1. Kengo Matsumoto
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Correspondence to Kengo Matsumoto.

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Matsumoto, K. Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras. Math. Z. 285, 121–141 (2017). https://doi.org/10.1007/s00209-016-1700-3

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