The strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras

TL;DR

This paper introduces and studies the concept of strong Morita equivalence for coactions of finite dimensional $C^*$-Hopf algebras on unital $C^*$-algebras, extending previous notions and analyzing their properties.

Contribution

It defines coactions on Hilbert $C^*$-bimodules, explores their properties, and shows that strong Morita equivalence preserves the Rohlin property for such coactions.

Findings

Strong Morita equivalence preserves the Rohlin property.

Introduces coactions on Hilbert $C^*$-bimodules of finite type.

Analyzes basic properties of coactions and their crossed products.

Abstract

Following Jansen and Waldmann, and Kajiwara and Watatani, we shall introduce notions of coactions of a finite dimensional $C^*$-Hopf algebra on a Hilbert $C^*$-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We shall investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras.