Coactions of a finite-dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and strong Morita equivalence
Volume 256 / 2021
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 show the following theorem: 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 $A’\cap (A\rtimes _{\rho , u}H)=\mathbb C 1$, 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 $\lambda ^0$.
