# Journal of Operator Theory

Volume 74, Issue 2, Fall 2015  pp. 329-369.

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

Authors:  Kazunori Kodaka (1) and Tamotsu Teruya (2)
Author institution: (1) Department of Mathematical Sciences, Faculty of Science, Ryukyu University, Nishihara-cho, Okinawa, 903-0213, Japan
(2) Faculty of Education, Gunma Univ., 4-2 Aramaki-machi, Maebashi City Gunma, 371-8510, Japan

Summary:  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 their basic properties. We shall give an example of a coaction of a finite dimensional $C^*$-Hopf algebra on a simple unital $C^*$-algebra, which has the above two properties and give the 1-cohomology and the 2-cohomology vanishing theorems for %twisted coactions of a finite dimensional $C^*$-Hopf algebra (twisted) coactions on a unital $C^*$-algebra. Furthermore, we shall show that if $\rho$ and $\sigma$, coactions of a finite dimensional $C^*$-Hopf algebra 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 $\sigma=(\alpha\otimes\id)\circ\rho\circ\alpha^{-1}$.

DOI: http://dx.doi.org/10.7900/jot.2014jul02.2029
Keywords:  $C^*$-algebras, finite dimensional $C^*$-Hopf algebras, approximately representable, the Rohlin property