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 the 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 $C^*$-alge\-bra of $A$ and $\rho ^s$ is the coaction of $H^0$ on $A^s$ induced by $\rho $. Furthermore, we shall show that $Pic _{H^0}^{\hat {\rho }}(A\rtimes _{\rho , u}H)$ 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 $A$.