The Structure of Nuclear C*-Algebras

本文通过研究核C~*-代数的同态、同构类型及其不变量以及C~*-代数的扩张来研究核C~*-代数。拓扑空间之间的连续映射在拓扑学中具有重要意义。在C~*-代数理论中，同态作为它们的非对易对偶具有重要的意义。这个项目的一个部分是确定从一个单位可分核C*-代数到另一个C*-代数的两个同态何时(稳定地)近似酉等价。判定两个C*-代数何时同构并刻画它们的不变量一直是C*-代数理论的中心问题。本项目的另一个部分是利用核可分C*-代数的K-理论对(简单)核可分C*-代数进行分类。这两个部分是密切相关的，都涉及到C*-代数的一般理论、KK-理论、自同构论、C*-代数扩张理论、群表示以及代数拓扑。工程、物理科学和社会科学中的许多重要问题都需要线性代数(或矩阵代数)的知识来解决。线性代数中的许多问题，特别是线性代数中的渐近和逼近问题，最好用无限维线性代数来描述。C*-代数的研究是对无限维线性代数的研究。然而，C*-代数的应用范围从线性代数、算子论、群表示、动力系统到模型量子场论和量子统计力学。对于C~*-代数理论的发展以及在其他领域的应用来说，对C~*-代数的结构有更好的了解是非常重要的。用最少的可能数据确定C*-代数的结构是这个项目的雄心勃勃的目标。

Abstract Lin The project is to study nuclear C*-algebras by studying their homomorphisms, isomorphic types and their invariants, and by studying extensions of C*-algebras. Continuous maps between topological spaces are of fundamental importance in topology. As their noncommutative counterparts, homomorphisms are of fundamental importance in the theory of C*-algebras. One part of the project is to determine when two homomorphisms from one unital separable nuclear C*-algebra to another C*-algebra are (stably) approximately unitarily equivalent. It is always in the center of the theory of C*-algebras to determine when two C*-algebras are isomorphic and to describe their invariants. Another part of the project is to classify (simple) nuclear separable C*-algebras via their K-theory. These two parts are closely related and both involve the general theory of C*-algebras, KK-theory, the theory of automorphisms, the C*-algebra extension theory, the group representations as well as algebraic topology. Many important and significant problems in engineering, the physical sciences, and social sciences require the knowledge of linear algebras (or matrix algebras) to solve them. Many problems in linear algebras, in particular asymptotic and approximation problems in linear algebras are best described in infinite dimensional linear algebras. The study of C*-algebras is the study of infinite dimensional linear algebras. However the applications of C*-algebras range from linear algebras, operator theory, group representations, dynamic systems, to model quantum field theory and quantum statistical mechanics. For the development of the theory of C*-algebras as well as for the purpose of applications to other areas, it is primarily important to have a better understanding of the structure of C*-algebras. It is the ambitious goal of this project to determine the structure of C*-algebras with fewest possible data.