Coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and the strong Morita equivalence

Abstract

Let $A$ and $B$ be unital $C^*$-algebras and let $H$ be a finite dimensional $C^*$-Hopf algebra. Let $H^0$ be its dual $C^*$-Hopf algebra. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. In this paper, we shall show the following theorem: We suppose that the unital inclusions $A\subset A\rtimes_{\rho, u}H$ and $B\subset B\rtimes_{\sigma, v}H$ are strongly Morita equivalent. If $A'\cap (A\rtimes_{\rho, u}H)=\BC1$, 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 $((\id_B \otimes\lambda^0 )\circ\sigma \, , \, (\id_B \otimes\lambda^0 \otimes\lambda^0 )(v))$ induced by $(\sigma, v)$ and $\lambda^0$.