The generalized Picard groups for finite dimensional $C^{\ast }$-Hopf algebra coactions on unital $C^{\ast }$-algebras

We shall generalize the notion of the strong Morita equivalence for coactions of a finite dimensional $C^{\ast }$-Hopf algebra on a unital $C^{\ast }$-algebra and define the Picard groups with respect to the generalized strong Morita equivalence. We call them the generalized Picard groups for coactions of a finite dimensional $C^{\ast }$-Hopf algebra on a unital $C^{\ast }$-algeba. We shall investigate basic properties of the generalized Picard groups and clarify the relation between the generalized Picard groups and the Picard groups for unital inclusions of unital $C^{\ast }$-algebras.