# 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

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