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2017 Equivariant Picard groups of$C^*$-algebras with finite dimensional$C^*$-Hopf algebra coactions
Kazunori Kodaka
Rocky Mountain J. Math. 47(5): 1565-1615 (2017). DOI: 10.1216/RMJ-2017-47-5-1565

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 $A$.

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Kazunori Kodaka. "Equivariant Picard groups of$C^*$-algebras with finite dimensional$C^*$-Hopf algebra coactions." Rocky Mountain J. Math. 47 (5) 1565 - 1615, 2017. https://doi.org/10.1216/RMJ-2017-47-5-1565

Information

Published: 2017
First available in Project Euclid: 22 September 2017

zbMATH: 06790026
MathSciNet: MR3705765
Digital Object Identifier: 10.1216/RMJ-2017-47-5-1565

Subjects:
Primary: 46L05
Secondary: 46L08

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 5 • 2017
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