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

TL;DR

This paper introduces the Rohlin property and approximate representability for coactions of finite dimensional $C^*$-Hopf algebras on unital $C^*$-algebras, establishing key properties, examples, and cohomology vanishing theorems.

Contribution

It defines new properties for coactions, provides examples, and proves cohomology vanishing theorems, advancing the understanding of coactions of finite dimensional $C^*$-Hopf algebras.

Findings

Established the Rohlin property for coactions.

Provided an example of an approximately representable coaction with the Rohlin property.

Proved 1- and 2-cohomology vanishing theorems for such coactions.

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 $\sigma=(\alpha\otimes\id)\circ\rho\circ\alpha^{-1}$.