The Picard groups for unital inclusions of unital $C^{\ast }$-algebras

We shall introduce the notion of the Picard group for an inclusion of $C^{\ast }$-algebras. We shall also study its basic properties and the relation between the Picard group for an inclusion of $C^{\ast }$-algebras and the ordinary Picard group. Furthermore, we shall give some examples of the Picard groups for unital inclusions of unital $C^{\ast }$-algebras.