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Transactions of the American Mathematical Society

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


Author: Kengo Matsumoto
Journal: Trans. Amer. Math. Soc. 370 (2018), 7011-7050
MSC (2010): Primary 46L55; Secondary 37B10
DOI: https://doi.org/10.1090/tran/7272
Published electronically: May 9, 2018
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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)$.


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Additional Information

Kengo Matsumoto
Affiliation: Department of Mathematics, Joetsu University of Education, Joetsu, 943-8512, Japan

DOI: https://doi.org/10.1090/tran/7272
Received by editor(s): October 26, 2016
Received by editor(s) in revised form: January 22, 2017, and February 18, 2017
Published electronically: May 9, 2018
Additional Notes: Ths work was supported by JSPS KAKENHI Grant Number 15K04896.
Article copyright: © Copyright 2018 American Mathematical Society

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