Abstract. Let $\CalH $ be a finite dimensional Hilbert space and $V$ a multiplicative unitary operator on $\CalHt $. Baaj and Skandalis showed that $V$ induces a finite dimensional $C^*$-Hopf algebra $H$ and its dual $C^*$-Hopf algebra $H^0$. Applying their results, Cuntz constructed a coaction $\lambda $ of $H$ on the Cuntz algebra $\CalO(\CalH )$, which is generated by $\CalH $. Let $\lambda |_C$ be its restriction to a canonical UHF-subalgebra $C$ of $\CalO(\CalH )$, which is a coaction of $H$ on $C$. In this paper, we shall show that $\lambda |_C$ is an approximately representable coaction of $H$ on $C$ with the Rohlin property.

DOI: 10.14232/actasm-016-024-9

AMS Subject Classification (1991): 46L05, 16T05

Keyword(s): $C^*$-algebras, finite dimensional $C^*$-Hopf algebras, approximately representable, multiplicative unitary, the Rohlin property

Received April 14, 2016, and in revised form September 1, 2016. (Registered under 24/2016.)