Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras

Matsumoto, Kengo

doi:10.1090/proc/13387

Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras
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by Kengo Matsumoto

Proc. Amer. Math. Soc. 145 (2017), 1131-1140

DOI: https://doi.org/10.1090/proc/13387

Published electronically: November 29, 2016

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Corrigendum: Proc. Amer. Math. Soc. (to appear).

Abstract:

We introduce a notion of uniformly continuous orbit equivalence as a subequivalence relation of continuous orbit equivalence of one-sided topological Markov shifts. It is described in terms of gauge actions on the associated Cuntz–Krieger algebras and continuous full groups of the Markov shifts.

References

Mike Boyle and David Handelman, Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math. 95 (1996), 169–210. MR 1418293, DOI 10.1007/BF02761039
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Masatoshi Enomoto, Masatoshi Fujii, and Yasuo Watatani, $K_{0}$-groups and classifications of Cuntz-Krieger algebras, Math. Japon. 26 (1981), no. 4, 443–460. MR 634920
Bruce P. Kitchens, Symbolic dynamics, Universitext, Springer-Verlag, Berlin, 1998. One-sided, two-sided and countable state Markov shifts. MR 1484730, DOI 10.1007/978-3-642-58822-8
Wolfgang Krieger, On dimension functions and topological Markov chains, Invent. Math. 56 (1980), no. 3, 239–250. MR 561973, DOI 10.1007/BF01390047
Wolfgang Krieger, On a dimension for a class of homeomorphism groups, Math. Ann. 252 (1979/80), no. 2, 87–95. MR 593623, DOI 10.1007/BF01420115
Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092, DOI 10.1017/CBO9780511626302
Kengo Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math. 246 (2010), no. 1, 199–225. MR 2645883, DOI 10.2140/pjm.2010.246.199
Kengo Matsumoto, $K$-groups of the full group actions on one-sided topological Markov shifts, Discrete Contin. Dyn. Syst. 33 (2013), no. 8, 3753–3765. MR 3021379, DOI 10.3934/dcds.2013.33.3753
Kengo Matsumoto, Classification of Cuntz-Krieger algebras by orbit equivalence of topological Markov shifts, Proc. Amer. Math. Soc. 141 (2013), no. 7, 2329–2342. MR 3043014, DOI 10.1090/S0002-9939-2013-11519-4
Kengo Matsumoto, Full groups of one-sided topological Markov shifts, Israel J. Math. 205 (2015), no. 1, 1–33. MR 3314581, DOI 10.1007/s11856-014-1134-8
Kengo Matsumoto, Strongly continuous orbit equivalence of one-sided topological Markov shifts, J. Operator Theory 74 (2015), no. 2, 457–483. MR 3431940, DOI 10.7900/jot.2014aug19.2063
K. Matsumoto, Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras, preprint, arXiv:1501.06965, to appear in Math. Z. DOI 10.1007/s00209-016-1700-3.
Kengo Matsumoto and Hiroki Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Kyoto J. Math. 54 (2014), no. 4, 863–877. MR 3276420, DOI 10.1215/21562261-2801849
Kengo Matsumoto and Hiroki Matui, Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions, Ergodic Theory Dynam. Systems 36 (2016), no. 5, 1557–1581. MR 3519423, DOI 10.1017/etds.2014.128
Hiroki Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. Lond. Math. Soc. (3) 104 (2012), no. 1, 27–56. MR 2876963, DOI 10.1112/plms/pdr029
Hiroki Matui, Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math. 705 (2015), 35–84. MR 3377390, DOI 10.1515/crelle-2013-0041
Yiu Tung Poon, A $K$-theoretic invariant for dynamical systems, Trans. Amer. Math. Soc. 311 (1989), no. 2, 515–533. MR 978367, DOI 10.1090/S0002-9947-1989-0978367-5
Şerban Strătilă and Dan Voiculescu, Representations of AF-algebras and of the group $U(\infty )$, Lecture Notes in Mathematics, Vol. 486, Springer-Verlag, Berlin-New York, 1975. MR 458188, DOI 10.1007/BFb0082276
M. Tomforde, The Graph Algebra Problem Page : List of Open Problems, http://www.math. uh.edu/tomforde/GraphAlgebraProblems/ListOfProblems.html.