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

Abstract

Let $A$ and $B$ be unital $C^{\ast }$-algebras and let $H$ be a finite dimensional $C^{\ast }$-Hopf algebra. Let $H^0$ be its dual $C^{\ast }$-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 $A\subset A{⋊}_{\rho ,u}H$ and $B\subset B{⋊}_{\sigma ,v}H$ are strongly Morita equivalent. If $A'\cap (A\rtimes_{ρ, u}H)=\BC1$, then there is a $C^{\ast }$-Hopf algebra automorphism ${\lambda }^0$ of $H^0$ such that the twisted coaction $(\rho ,u)$ is strongly Morita equivalent to the twisted coaction $((\id_B \otimesλ^0 )\circσ\, , \, (\id_B \otimesλ^0 \otimesλ^0 )(v))$ induced by $(\sigma ,v)$ and ${\lambda }^0$.

17 pages

References in corpus (1)

• The strong Morita equivalence for inclusions of $C^{\ast }$-algebras and conditional expectations for equivalence bimodules

Cited by in corpus (1)

• The Picard groups of unital inclusions of unital $C^{\ast }$-algebras induced by involutive equivalence bimodules