Simple Amenable C*-algebras and K-theory

在量子力学中，或在微观物理世界中，一个可观测量可以用希尔伯特空间上的自伴算子来建模。一个这样的算子系统形成一个C *-代数。在有限维情形下，这些算子只是矩阵。在更一般的情况下，这些算子可以用无穷矩阵或无穷维希尔伯特空间上的矩阵来表示。在量子力学的数学公式中，一对代表观测量的自伴算符通常是非对易的，这对应于海森堡测不准原理。C *-代数的研究通常被看作是拓扑空间上连续函数代数的非交换类似物的研究。C~*-代数来自于不同的科学领域，其研究有着广泛的应用。我们知道，不同应用中的不同表现的C *-代数可以是相同的。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。从应用的角度和理论的角度来看，从C *-代数的外观确定其结构是极其重要的。这个项目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，人们可以通过研究可计算数据来理解C *-代数的结构。本项目主要研究基于K-理论的顺从C *-代数的不变量分类及其应用。这个项目的主要部分是寻找一个广泛的分类的非单位简单顺从C *-代数。该策略的一部分是创建新的技术工具来处理非紧空间的非对易版本。目的是建立一类新的存在性定理，以及非单位单C *-代数的渐近酉等价的K-理论刻画。一个密切相关的主题是证明某些C *-代数满足泛系数定理。该项目的研究也将提供一个新的通道，大大简化了C *-代数理论应用于其他各种相关研究领域的方式，特别是非交换同伦理论和动力系统的研究。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，称之为渐近共轭，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.