The Rohlin property for coactions of finite dimensional $C^*$-Hopf algebras on unital $C^*$-algebras

Abstract: We shall introduce the approximate representability and the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and discuss some basic properties of approximately representable coactions and coactions with the Rohlin property of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra. Also, we shall give an example of an approximately representable coaction of a finite dimensional $C^*$-Hopf algebra on a simple unital $C^*$-algebra which has also the Rohlin property and we shall give the 1-cohomology vanishing theorem for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and the 2-cohomology vanishing theorem for twisted coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra. Furthermore, we shall introduce the notion of the approximately unitary equivalence of coactions of a finite dimensional $C^*$-Hopf algebra $H$ on a unital $C^*$-algebra $A$ and show that if $\rho$ and $\sigma$, coactions of $H$ on a separable unital $C^*$-algebra $A$, which have the Rohlin property, are approximately unitarily equivalent, then there is an approximately inner automorphism $\alpha$ on $A$ such that $\sigma=(\alpha\otimes\id)\circ\rho\circ\alpha^{-1}$.