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

TL;DR

This paper characterizes when inclusions of crossed product $C^*$-algebras induced by equivalence bimodules are strongly Morita equivalent, and computes their Picard groups under specific conditions.

Contribution

It provides a necessary and sufficient condition for strong Morita equivalence of such inclusions and calculates their Picard groups.

Findings

Strong Morita equivalence characterized by bimodule isomorphisms.

Explicit computation of the Picard group for certain $C^*$-algebra inclusions.

Conditions involving dual bimodules are key to the equivalence.

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 $A' \cap M(A\rtimes_X \mathbb{Z})=\mathbb{C} 1$.