The generalized Picard groups for finite dimensional $C^*$-Hopf algebra coactions on unital $C^*$-algebras

TL;DR

This paper introduces a generalized notion of Picard groups for coactions of finite dimensional $C^*$-Hopf algebras on unital $C^*$-algebras, extending the concept of strong Morita equivalence and exploring their properties.

Contribution

It defines and studies the properties of generalized Picard groups for $C^*$-Hopf algebra coactions, linking them to existing Picard groups for unital inclusions.

Findings

Established basic properties of the generalized Picard groups.

Clarified the relationship between generalized Picard groups and Picard groups for unital inclusions.

Extended the framework of Morita equivalence for coactions of $C^*$-Hopf algebras.

Abstract

We shall generalize the notion of the strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-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 $C^*$-Hopf algebra on a unital $C^*$-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 $C^*$-algebras.