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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
MathSciNet review: 3841841
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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
MR Author ID: 205406

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