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College of Natural Sciences (자연과학대학)
Dept. of Mathematical Sciences (수리과학부)
Theses (Ph.D. / Sc.D._수리과학부)

Crossed products of Cuntz-Pimsner algebras by coactions of Hopf $C^*$-algebras*호프 $C^*$-대수의 쌍대작용에 의한 쿤쯔-핌스너 대수의 교차곱*

- Authors
- 김동운

- Advisor
- 정자아

- Major
- 자연과학대학 수리과학부

- Issue Date
- 2015-02

- Publisher
- 서울대학교 대학원

- Keywords
- $C^*$-correspondence; Cuntz-Pimsner algebra; multiplier correspondence; Hopf $C^*$-algebra; coaction; reduced crossed product

- Description
- 학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2015. 2. 정자아.

- Abstract
- Unifying two notions of an action and coaction of a locally compact group on a $C^*$-cor\-re\-spond\-ence we introduce a coaction $(\sigma,\delta)$ of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$. We show that this coaction naturally induces a coaction $\zeta$ of $S$ on the associated Cuntz-Pimsner algebra $\mathcal{O}_X$ under the weak $\delta$-invariancy for the ideal $J_X$. When the Hopf $C^*$-algebra $S$ is a reduced Hopf $C^*$-algebra of a well-behaved multiplicative unitary, we construct from the coaction $(\sigma,\delta)$ a $C^*$-cor\-re\-spond\-ence $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$, and show that it has a representation on the reduced crossed product $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ by the induced coaction $\zeta$. If this representation is covariant, particularly if either the ideal $J_{X\rtimes_\sigma\widehat{S}}$ of $A\rtimes_\delta\widehat{S}$ is generated by the canonical image of $J_X$ in $M(A\rtimes_\delta\widehat{S})$ or the left action on $X$ by $A$ is injective, the $C^*$-algebra $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ is shown to be isomorphic to the Cuntz-Pimsner algebra $\mathcal{O}_{X\rtimes_\sigma\widehat{S}}$ associated to $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$. Under the covariance assumption, our results extend the isomorphism result known for actions of amenable groups to arbitrary locally compact groups. Also, the Cuntz-Pimsner covariance condition which was assumed for the same isomorphism result concerning group coactions is shown to be redundant.

- Language
- English

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