호프 $C^*$-대수의 쌍대작용에 의한 쿤쯔-핌스너 대수의 교차곱

Abstract

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