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The Picard groups of inclusions of $C^*$-algebras induced by equivalence bimodules

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  06 July 2021

Kazunori Kodaka
Kazunori Kodaka*
Affiliation:
Department of Mathematical Sciences, Faculty of Science, Ryukyu University, Nishihara-cho, Okinawa903-0213, Japan
*
e-mail: [email protected]
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Abstract

For two $\sigma $ -unital $C^*$ -algebras, we consider two equivalence bimodules over them, respectively. Then, by taking the crossed products by the equivalence bimodules, we get two inclusions of $C^*$ -algebras. Furthermore, we suppose that one of the inclusions of $C^*$ -algebras is irreducible, that is, the relative commutant of one of the $\sigma $ -unital $C^*$ -algebras in the multiplier $C^*$ -algebra of the crossed product is trivial. We will give a sufficient and necessary condition that the two inclusions are strongly Morita equivalent. Applying this result, we will compute the Picard group of a unital inclusion of unital $C^*$ -algebras induced by an equivalence bimodule over the unital $C^*$ -algebra under the assumption that the unital inclusion of unital $C^*$ -algebras is irreducible.

Keywords

Crossed productsequivalence bimodulesinclusions of C*-algebrasstrong Morita equivalencePicard groups

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L08: $C^*$-modules
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 743 - 758
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Copyright
© Canadian Mathematical Society 2021

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References

Abadie, B., Eilers, S., and Exel, R., Morita equivalence for crossed products by Hilbert C*-bimodules. Trans. Amer. Math. Soc. 350(1998), 3043–3054.CrossRefGoogle Scholar
Bédos, E. and Conti, R., On discrete twisted C*-dynamical systems, Hilbert C*-modules and regularity. Münster J. Math. 5(2012), 183–208.Google Scholar
Brown, L. G., Green, P., and Rieffel, M. A., Stable isomorphism and strong Morita equivalence of C*-algebras. Pacific J. Math. 71(1977), 349–363.CrossRefGoogle Scholar
Brown, L. G., Mingo, J., and Shen, N.-T., Quasi-multipliers and embeddings of Hilbert C*-bimodules. Canad. J. Math. 46(1994), 1150–1174.10.4153/CJM-1994-065-5CrossRefGoogle Scholar
Jensen, K. K. and Thomsen, K., Elements of KK-theory, Birkhuser Birkhäuser, 1991.10.1007/978-1-4612-0449-7CrossRefGoogle Scholar
Kodaka, K., The Picard groups for unital inclusions of unital C*-algebras. Acta Sci. Math. (Szeged) 86(2020), 183–207.CrossRefGoogle Scholar
Kodaka, K., Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital C*-algebras. Math. Bohem., to appear.Google Scholar
Kodaka, K., Strong Morita equivalence for inclusions of C*-algebras induced by twisted actions of a countable discrete group. Math. Scand., to appear.Google Scholar
Kodaka, K. and Teruya, T., Involutive equivalence bimodules and inclusions of C*-algebras with Watatani index 2. J. Operator Theory 57(2007), 3–18.Google Scholar
Kodaka, K. and Teruya, T., The strong Morita equivalence for inclusions of C*-algebras and conditional expectations for equivalence bimodules. J. Aust. Math. Soc. 105(2018), 103–144.CrossRefGoogle Scholar
Kodaka, K. and Teruya, T., Coactions of a finite dimensional C*-Hopf algebra on unital C*-algebras, unital inclusions of unital C*-algebras and the strong Morita equivalence. Studia Math. 256(2021), 147–167.CrossRefGoogle Scholar