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

TL;DR

This paper computes the Picard group of a specific unital inclusion of $C^*$-algebras induced by an involutive equivalence bimodule, under the condition that the relative commutant is trivial.

Contribution

It provides a formula for the Picard group of unital inclusions of $C^*$-algebras induced by involutive equivalence bimodules, assuming a trivial relative commutant.

Findings

Explicit computation of the Picard group for the given inclusion.

Conditions under which the Picard group can be determined.

Extension of known results to involutive bimodule cases.

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 $A\subset C_X$.