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

Abstract: Let $A$ be a unital $C^*$-algebra and $X$ an invulutive $A-A$-equivalence bimodule. Let $A\subset C_X$ be the unital inclusion of unital $C^*$-algebras induced by $X$. We suppose that $A' \cap C_X =\mathbf{C} 1$. We shall compute the Picard group of the unital inclusion $A\subset C_X$.