Equivariant Picard groups of $C^*$-algebras with finite dimensional $C^*$-Hopf algebra coactions

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