The Picard groups of inclusions of -algebras induced by equivalence bimodules

Abstract

AbstractFor 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.