Definition 2.1]. If $A'\cap (A\rtimes_{\rho, u}H)={\mathbf C}1$, then there is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that the twisted coaction $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $(({{\rm{id}}}_B \otimes\lambda^0 )\circ\sigma \, , \, ({{\rm{id}}}_B \otimes\lambda^0 \otimes\lambda^0 )(v))$ induced by $(\sigma, v)$ and $\lambda^0$.

For a unital $C^*$-algebra $A$, let $M_n (A)$ be the $n\times n$-matrix algebra over $A$ and $I_n$ denotes the unit element in $M_n (A)$. We identify $M_n (A)$ with $A\otimes M_n ({\mathbf C})$.

Let $A$ and $B$ be $C^*$-algebras and $X$ an $A-B$-bimodule. We denote its left $A$-action and right $B$-action on $X$ by $a\cdot x$ and $x\cdot b$ for any $a\in A$, $b\in B$, $x\in X$, respectively. Also, we denote by $\widetilde{X}$ the dual $B-A$-bimodule of $X$ and we denote by $\widetilde{x}$ the element in $\widetilde{X}$ induced by $x\in X$.

Preliminaries {#sec:pre}
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Let $H$ be a finite dimensional $C^*$-Hopf algebra. We denote its comultipication, counit and antipode by $\Delta$, $\epsilon$ and $S$, respectively. We shall use Sweedler’s notation, $\Delta(h)=h_{(1)}\otimes h_{(2)}$ for any $h\in H$ which suppresses a possible summation when we write comultiplications. We denote by $N$ the dimension of $H$. Let $H^0$ be the dual $C^*$-Hopf algebra of $H$. 