Simple Amenable C*-algebras and K-theory

在量子力学中，或在微观物理世界中，可观测对象可以由希尔伯特空间上的自伴算子来建模。这样的算符系统形成了一个C*-代数。在有限维的情况下，这些算子只是矩阵。在更一般的情况下，这些算子可以用无限矩阵或无限维希尔伯特空间上的矩阵来表示。在量子力学的数学表述中，一对表示可观测的自伴算符通常是非对易的，这与海森堡测不准原理相对应。C*-代数的研究通常被看作是对拓扑空间上的连续函数代数的非交换类似的研究。有许多C*-代数来自不同的科学领域，对C*-代数的研究有着广泛的应用。众所周知，具有不同外观的C*-代数在不同的应用中可以是相同的。另一方面，来自同一来源的C*-代数可能看起来相同，但具有本质上不同的结构。无论是从应用的角度还是从理论的角度来看，根据C*-代数的外观来确定C*-代数的结构是非常重要的。这个项目是试图使用少量的可计算数据来完全确定某些类中的C*-代数。换句话说，人们可以研究可计算的数据来理解C*-代数的结构。本课题是利用K-理论对依从性C~*-代数进行分类的研究，涉及的不变量及其分类的应用。这个项目的主要部分是寻找非单位简单服从C*-代数的一个广泛的分类。战略的一部分是创造新的技术工具来处理非交换版本的非紧凑空间。目的是建立非酉单C~*-代数的一种新的存在定理和渐近酉等价性的K-理论刻画。一个密切相关的话题是验证某些C*-代数是否满足泛系数定理。本课题的研究也将大大简化C*-代数理论应用于其他相关研究领域，特别是非对易同伦理论和动力系统研究的新途径。例如，利用这一广义分类理论，研究了紧度量空间上极小同胚之间的一种新的关系，称为渐近共轭，并最终利用K-理论数据刻画了这种关系。

In quantum mechanics, or in the microscopical physical world, an observable may be modeled by a self-adjoint operator on a Hilbert space. A system of such operators forms a C*-algebra. In the finite dimensional cases, these operators are just matrices. In more general cases, these operators may be represented by infinite matrices, or matrices on infinite dimensional Hilbert space. In the mathematical formulation of quantum mechanics, a pair of self-adjoint operators representing observables are typically non-commutative which corresponds to the Heisenberg uncertainty principle. The study of C*-algebra is often viewed as the study of non-commutative analogues of the algebra of continuous functions on a topological space. There are many C*-algebras that come from different fields of sciences and the study of C*-algebras has variety of applications. It is known that C*-algebras with different appearance from different applications can be the same. On the other hand, C*-algebras from the same source may look the same but have essentially different structure. For application purposes as well as the theoretical point of view, it is extremely important to determine the structure of C*-algebras from their appearance. This project is an attempt to use a small number of computable data to completely determine C*-algebras in certain classes. In other words one may study the computable data to understand the structure of C*-algebras. This project is a study of classification of amenable C*-algebras using K-theory related invariants and applications of the classification. The main part of this project is to search for a broad classification of non-unital simple amenable C*-algebras. Part of the strategy is to create new technical tools to deal with non-commutative version of non-compact spaces. The goal is to establishe a new type of existence theorem as well as a K-theoretic characterization of asymptotic unitary equivalence for non-unital simple C*-algebras. A closely related topic is to verify certain C*-algebras satisfy the Universal Coefficient Theorem. The research of this project will also provide a new passage to greatly simplify the way in which the theory of C*-algebras is applied to various other related research areas, notably, to non-commutative homotopy theory and the study of dynamical systems. For example, it is proposed, using this broad classification theory, to study a new relation among minimal homeomorphisms on a compact metric space which is called asymptotic conjugacy, and ultimately characterize this relation via K-theoretical data.