Equivariant Picard groups of $C^{\ast }$-algebras with finite dimensional $C^{\ast }$-Hopf algebra coactions

Let $A$ be a $C^{\ast }$-algebra and $H$ a finite dimensional $C^{\ast }$-Hopf algebra with its dual $C^{\ast }$-Hopf algebra $H^0$. Let $(\rho ,u)$ be a twisted coaction of $H^0$ on $A$. We shall define the $(\rho ,u,H)$-equivariant Picard group of $A$, which is denoted by ${\text{Pic}}_H^{\rho ,u}(A)$, and discuss basic properties of ${\text{Pic}}_H^{\rho ,u}(A)$. Also, we suppose that $(\rho ,u)$ is the coaction of $H^0$ on the unital $C^{\ast }$-algebra $A$, that is, $u=1\otimes 1^0$. We investigate the relation between $\text{Pic}(A^s)$, the ordinary Picard group of $A^s$ and ${\text{Pic}}_H^{{\rho }^s}(A^s)$ where $A^s$ is the stable $C^{\ast }$-algebra of $A$ and ${\rho }^s$ is the coaction of $H^0$ on $A^s$ induced by $\rho$. Furthermore, we shall show that $\Pic_{H^0}^{\widehat{\rho}}(A\rtimes_{\rho, u}H)$isisomorphicto$\Pic_H^{\rho, u}(A)$,where$\widehat{\rho}$ is the dual coaction of $H$ on the twisted crossed product $A{⋊}_{\rho ,u}H$ of $A$ by the twisted coaction $(\rho ,u)$ of $H^0$ on $A$.