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 \(A\).


46L05 General theory of \(C^*\)-algebras
46L08 \(C^*\)-modules
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