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

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dc.contributor.author김동운-
dc.date.accessioned2017-07-14T00:41:27Z-
dc.date.available2017-07-14T00:41:27Z-
dc.date.issued2015-02-
dc.identifier.other000000025319-
dc.identifier.urihttps://hdl.handle.net/10371/121291-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2015. 2. 정자아.-
dc.description.abstractUnifying 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.-
dc.description.tableofcontentsAbstract
1. Introduction
2. Preliminaries
2.1. $C^*$-correspondences
2.2. Multiplier correspondences
2.3. Tensor product correspondences
2.4. Cuntz-Pimsner algebras
2.5. $C$-multiplier correspondences
2.6. Reduced and dual reduced Hopf $C^*$-algebras
2.7. Reduced crossed products $A\rtimes\widehat{S}$
3. Coactions of Hopf $C^*$-algebras on $C^*$-correspondences
3.1. The extensions $(\overline{k_X\otimes{\rm id}},\overline{k_A\otimes{\rm id}})$
3.2. Coactions on $C^*$-correspondences and their induced coactions
4. Reduced crossed product correspondences
4.1. Baaj-Skandalis type lemma for $C^*$-correspondences
4.2. Reduced crossed product correspondences $(X\rtimes\widehat{S},A\rtimes\widehat{S})$
5. Reduced crossed products
5.1. Representations of $(X\rtimes\widehat{S},A\rtimes\widehat{S})$ on $\mathcal{O}_X\rtimes\widehat{S}$
5.2. An isomorphism between $\mathcal{O}_X\rtimes\widehat{S}$ and ${O}_{X\rtimes\widehat{S}}$
6. Examples
6.1. Coactions on crossed products by $\mathbb{Z}$
6.2. Coactions on directed graph $C^*$-algebras
6.2.1. Labelings and coactions on graph $C^*$-algebras
6.2.2. Coactions on finite graphs
Appendix A. Coactions of $C_0(G)$ on $C^*$-correspondences
A.1. Akemann-Pedersen-Tomiyama type theorem for $C^*$-correspondences
A.2. One-to-one correspondence between $G$-actions and $C_0(G)$-coactions
Appendix B. $C^*$-correspondences $(X\rtimes\widehat{S}_{\widehat{W}_G},A\rtimes\widehat{S}_{\widehat{W}_G})$
B.1. $C^*$-correspondences $(\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H}),\mathcal{L}_A(A\otimes\mathcal{H}))$
B.2. Crossed product correspondences $(X\rtimes_r G,A\rtimes_r G)$
Abstract (in Korean)
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dc.formatapplication/pdf-
dc.format.extent2524687 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subject$C^*$-correspondence-
dc.subjectCuntz-Pimsner algebra-
dc.subjectmultiplier correspondence-
dc.subjectHopf $C^*$-algebra-
dc.subjectcoaction-
dc.subjectreduced crossed product-
dc.subject.ddc510-
dc.titleCrossed products of Cuntz-Pimsner algebras by coactions of Hopf $C^*$-algebras-
dc.title.alternative호프 $C^*$-대수의 쌍대작용에 의한 쿤쯔-핌스너 대수의 교차곱-
dc.typeThesis-
dc.description.degreeDoctor-
dc.citation.pagesiii, 96-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2015-02-
Appears in Collections:
College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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