Relative Morita equivalence of Cuntz–Krieger algebras and flow equivalence of topological Markov shifts

Matsumoto, Kengo

doi:10.1090/tran/7272

Relative Morita equivalence of Cuntz–Krieger algebras and flow equivalence of topological Markov shifts
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Trans. Amer. Math. Soc. 370 (2018), 7011-7050 Request permission

Abstract:

We will introduce a relative version of imprimitivity bimodule and a relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal {A}, \mathcal {D})$ such that $\mathcal {D}$ is a $C^*$-subalgebra of $\mathcal {A}$ satisfying certain conditions. We will then prove that two pairs $(\mathcal {A}_1, \mathcal {D}_1)$ and $(\mathcal {A}_2, \mathcal {D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particular, for two pairs $(\mathcal {O}_A, \mathcal {D}_A)$ and $(\mathcal {O}_B, \mathcal {D}_B)$ of Cuntz–Krieger algebras with their canonical masas, they are relatively Morita equivalent if and only if their underlying two-sided topological Markov shifts $(\overline {X}_A,\bar {\sigma }_A)$ and $(\overline {X}_B,\bar {\sigma }_B)$ are flow equivalent. We also introduce a relative version of the Picard group ${\operatorname {Pic}}(\mathcal {A}, \mathcal {D})$ for the pair $(\mathcal {A}, \mathcal {D})$ of $C^*$-algebras and study them for the Cuntz–Krieger pair $(\mathcal {O}_A, \mathcal {D}_A)$.

References

T. Bates, Application of the gauge-invariant uniqueness theorem for the Cuntz–Krieger algebras of directed graphs, Bull. Austral. Math. Soc. 65 (2002), 57–67.
Teresa Bates and David Pask, Flow equivalence of graph algebras, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 367–382. MR 2054048, DOI 10.1017/S0143385703000348
Rufus Bowen and John Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73–92. MR 458492, DOI 10.2307/1971159
Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348. MR 454645, DOI 10.2140/pjm.1977.71.335
Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928, DOI 10.2140/pjm.1977.71.349
Nathan Brownlowe, Toke Meier Carlsen, and Michael F. Whittaker, Graph algebras and orbit equivalence, Ergodic Theory Dynam. Systems 37 (2017), no. 2, 389–417. MR 3614030, DOI 10.1017/etds.2015.52
Robert C. Busby, Double centralizers and extensions of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99. MR 225175, DOI 10.1090/S0002-9947-1968-0225175-5
T. M. Carlsen, S. Eilers, E. Ortega and G. Restorff, Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids, preprint, arXiv:1610.09945 [math.DS].
Toke Meier Carlsen, Efren Ruiz, and Aidan Sims, Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $C^*$-algebras and Leavitt path algebras, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1581–1592. MR 3601549, DOI 10.1090/proc/13321
Roberto Conti, Jeong Hee Hong, and Wojciech Szymański, The Weyl group of the Cuntz algebra, Adv. Math. 231 (2012), no. 6, 3147–3161. MR 2980494, DOI 10.1016/j.aim.2012.09.003
Roberto Conti, Jeong Hee Hong, and Wojciech Szymański, On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 2, 269–279. MR 3327955, DOI 10.1017/S0308210513001364
Joachim Cuntz, $K$-theory for certain $C^{\ast }$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. MR 604046, DOI 10.2307/1971137
J. Cuntz, A class of $C^{\ast }$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C^{\ast }$-algebras, Invent. Math. 63 (1981), no. 1, 25–40. MR 608527, DOI 10.1007/BF01389192
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
John Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 53–66. MR 758893, DOI 10.1017/S0143385700002261
Danrung Huang, Flow equivalence of reducible shifts of finite type, Ergodic Theory Dynam. Systems 14 (1994), no. 4, 695–720. MR 1304139, DOI 10.1017/S0143385700008129
Evgenios T. A. Kakariadis and Elias G. Katsoulis, Contributions to the theory of $\mathrm {C}^*$-correspondences with applications to multivariable dynamics, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6605–6630. MR 2958949, DOI 10.1090/S0002-9947-2012-05627-3
Evgenios T. A. Kakariadis and Elias G. Katsoulis, C*-algebras and equivalences for C*-correspondences, J. Funct. Anal. 266 (2014), no. 2, 956–988. MR 3132734, DOI 10.1016/j.jfa.2013.09.018
Tsuyoshi Kajiwara and Yasuo Watatani, Jones index theory by Hilbert $C^*$-bimodules and $K$-theory, Trans. Amer. Math. Soc. 352 (2000), no. 8, 3429–3472. MR 1624182, DOI 10.1090/S0002-9947-00-02392-8
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
Kazunori Kodaka, Picard groups of irrational rotation $C^*$-algebras, J. London Math. Soc. (2) 56 (1997), no. 1, 179–188. MR 1462834, DOI 10.1112/S0024610797005243
Kazunori Kodaka, Full projections, equivalence bimodules and automorphisms of stable algebras of unital $C^*$-algebras, J. Operator Theory 37 (1997), no. 2, 357–369. MR 1452282
Alexander Kumjian, On $C^\ast$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR 854149, DOI 10.4153/CJM-1986-048-0
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, Strong shift equivalence of symbolic dynamical systems and Morita equivalence of $C^\ast$-algebras, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 199–215. MR 2041268, DOI 10.1017/S014338570300018X
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, 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, Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz-Krieger algebras, Math. Z. 285 (2017), no. 1-2, 121–141. MR 3598806, DOI 10.1007/s00209-016-1700-3
Kengo Matsumoto, Topological conjugacy of topological Markov shifts and Cuntz-Krieger algebras, Doc. Math. 22 (2017), 873–915. MR 3665401
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
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
Paul S. Muhly, David Pask, and Mark Tomforde, Strong shift equivalence of $C^*$-correspondences, Israel J. Math. 167 (2008), 315–346. MR 2448028, DOI 10.1007/s11856-008-1051-9
Bill Parry and Dennis Sullivan, A topological invariant of flows on $1$-dimensional spaces, Topology 14 (1975), no. 4, 297–299. MR 405385, DOI 10.1016/0040-9383(75)90012-9
William L. Paschke, Inner product modules over $B^{\ast }$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. MR 355613, DOI 10.1090/S0002-9947-1973-0355613-0
Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. MR 1634408, DOI 10.1090/surv/060
Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017
Jean Renault, Examples of masas in $C^*$-algebras, Operator structures and dynamical systems, Contemp. Math., vol. 503, Amer. Math. Soc., Providence, RI, 2009, pp. 259–265. MR 2590626, DOI 10.1090/conm/503/09903
Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1
Mikael Rørdam, Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58. MR 1340839, DOI 10.1007/BF00965458
Mark Tomforde, Strong shift equivalence in the $C^*$-algebraic setting: graphs and $C^*$-correspondences, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 221–230. MR 2277213, DOI 10.1090/conm/414/07811
N. E. Wegge-Olsen, $K$-theory and $C^*$-algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. A friendly approach. MR 1222415
Yasuo Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117. MR 996807, DOI 10.1090/memo/0424
R. F. Williams, Classification of subshifts of finite type, Ann. of Math. (2) 98 (1973), 120–153; errata, ibid. (2) 99 (1974), 380–381. MR 331436, DOI 10.2307/1970908