## Rocky Mountain Journal of Mathematics

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

Kazunori Kodaka

#### 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 $A$.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 5 (2017), 1565-1615.

Dates
First available in Project Euclid: 22 September 2017

https://projecteuclid.org/euclid.rmjm/1506045625

Digital Object Identifier
doi:10.1216/RMJ-2017-47-5-1565

Mathematical Reviews number (MathSciNet)
MR3705765

Zentralblatt MATH identifier
06790026

Subjects
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L08: $C^*$-modules

#### Citation

Kodaka, Kazunori. Equivariant Picard groups of$C^*$-algebras with finite dimensional$C^*$-Hopf algebra coactions. Rocky Mountain J. Math. 47 (2017), no. 5, 1565--1615. doi:10.1216/RMJ-2017-47-5-1565. https://projecteuclid.org/euclid.rmjm/1506045625

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