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

2019/11/18 by Kodaka, Kazunori
#46L05 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1911.07477

Abstract

Let A and B be σ-unital C^*-algebras and X and Y an A-A-equivalence bimodule and a B-B-equivalence bimodule, respectively. Also, let A\rtimes_X ℤ and B\rtimes_Y ℤ be the crossed products of A and B by X and Y, respectively. Furthermore, let A⊂ A\rtimes_X ℤ and B⊂ B\rtimes_Y ℤ be the inclusions of C^*-algebras induced by X and Y, respectively. We suppose that A' ∩ M(A\rtimes_X ℤ)=ℂ 1. In this paper we shall show that the inclusions A⊂ A\rtimes_X ℤ and B⊂ B\rtimes_Y ℤ are strongly Morita equivalent if and only if there is an A-B-equivalence bimodule M such that Y≅ \widetildeM⊗_A X ⊗_A M or \widetildeY≅ \widetildeM⊗_A X ⊗_A M as B-B-equivalence bimodules, where \widetildeM and \widetildeY are the dual B-A-equivalence bimodule and the dual B-B-equivalence bimodule of M and Y, respectively. Applying this result, we shall compute the Picard group of the inclusion A⊂ A\rtimes_X ℤ under the assumption that A' ∩ M(A\rtimes_X ℤ)=ℂ 1.

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