---
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 $A''\cap (A\rtimes_{\rho, u}H)={\mathbf C}1$, 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 $\lambda^0$.'
address:
- 'Department of Mathematical Sciences, Faculty of Science, Ryukyu University, Nishihara-cho, Okinawa, 903-0213, Japan'
- 'Faculty of Education, Gunma University, 4-2 Aramaki-machi, Maebashi City, Gunma, 371-8510, Japan'
- '*[E-mail address]{}: [kodaka@math.u-ryukyu.ac.jp]{}*'
- '*[E-mail address]{}: [teruya@gunma-u.ac.jp]{}*'
author:
- Kazunori Kodaka and Tamotsu Teruya
title: 'Coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and the strong Morita equivalence'
---

Introduction {#sec:intro}
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In the previous papers [@KT3:equivalence] and [@KT4:morita], we discussed the strong Morita equivalences for twisted coactions of a finite 