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

Abstract

Let $A$ and $B$ be $\sigma$-unital $C^*$-algebras and $X$ and $Y$ an $A-A$-equivalence bimodule and a $B-B$-equivalence bimodule, respectively. Also, let $A\rtimes_X \mathbb{Z}$ and $B\rtimes_Y \mathbb{Z}$ be the crossed products of $A$ and $B$ by $X$ and $Y$, respectively. Furthermore, let $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_Y \mathbb{Z}$ be the inclusions of $C^*$-algebras induced by $X$ and $Y$, respectively. We suppose that $A' \cap M(A\rtimes_X \mathbb{Z})=\mathbb{C} 1$. In this paper we shall show that the inclusions $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_Y \mathbb{Z}$ are strongly Morita equivalent if and only if there is an $A-B$-equivalence bimodule $M$ such that $Y\cong \widetilde{M}\otimes_A X \otimes_A M$ or $\widetilde{Y}\cong \widetilde{M}\otimes_A X \otimes_A M$ as $B-B$-equivalence bimodules, where $\widetilde{M}$ and $\widetilde{Y}$ 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 $A\subset A\rtimes_X \mathbb{Z}$ under the assumption that $A' \cap M(A\rtimes_X \mathbb{Z})=\mathbb{C} 1$.