Geodesic
The Picard groups of inclusions of $C^*$-algebras induced by equivalence bimodules
Canadian mathematical bulletin, Tome 65 (2022) no. 3, pp. 743-758

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DOI

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.
DOI : 10.4153/S0008439521000497
Mots-clés : Crossed products, equivalence bimodules, inclusions of C*-algebras, strong Morita equivalence, Picard groups 
Kodaka, Kazunori. The Picard groups of inclusions of $C^*$-algebras induced by equivalence bimodules. Canadian mathematical bulletin, Tome 65 (2022) no. 3, pp. 743-758. doi: 10.4153/S0008439521000497
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     author = {Kodaka, Kazunori},
     title = {The {Picard} groups of inclusions of $C^*$-algebras induced by equivalence bimodules},
     journal = {Canadian mathematical bulletin},
     pages = {743--758},
     year = {2022},
     volume = {65},
     number = {3},
     doi = {10.4153/S0008439521000497},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000497/}
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