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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, Okinawa 903-0213, Japan
*
e-mail: kodaka@math.u-ryukyu.ac.jp
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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

Information

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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