## Equivariant Picard groups of $$C^\ast$$-algebras with finite dimensional $$C^\ast$$-Hopf algebra coactions.(English)Zbl 1387.46044

Summary: 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 $$\mathrm{Pic}_H^{\rho,u}(A)$$, and discuss the basic properties of $$\mathrm{Pic}_H^{\rho,u}(A)$$. 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 $$\mathrm{Pic}(A^s)$$, the ordinary Picard group of $$A^s$$, and $$\mathrm{Pic}_H^{\rho^{s}}(A^s)$$, 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 $$\mathrm{Pic}_{H^0}^{\hat{\rho}}(A\rtimes_{\rho,u}H)$$ is isomorphic to $$\mathrm{Pic}_H^{\rho,u}(A)$$, 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 $$A$$.

### MSC:

 46L05 General theory of $$C^*$$-algebras 46L08 $$C^*$$-modules
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