Simple Amenable C*-algebras and K-theory

在量子力学中，或者在微观物理世界中，一个可观测对象可以用希尔伯特空间上的自伴随算子来建模。由这些算子组成的系统构成了C*-代数。在有限维的情况下，这些算子就是矩阵。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。在量子力学的数学公式中，一对表示可观测值的自伴随算子是典型的非交换的，这与海森堡的测不准原理相对应。C*-代数的研究通常被看作是对拓扑空间上连续函数代数的非交换类似物的研究。有许多C*-代数来自不同的科学领域，对C*-代数的研究有各种各样的应用。已知在不同应用中具有不同外观的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。从应用和理论的角度来看，从C*-代数的外观来确定它们的结构是极其重要的。本项目尝试使用少量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。本课题研究了可调C*-代数的k -不变量分类及其应用。本项目的主要部分是寻找非单一简单可服从C*代数的广义分类。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交换版本。目的是建立非幺正简单C*-代数的一类新的存在性定理和渐近幺正等价的k理论刻画。一个密切相关的课题是验证某些C*-代数满足普适系数定理。本项目的研究也将为极大地简化C*-代数理论应用于其他相关研究领域，特别是非交换同伦理论和动力系统的研究提供新的途径。例如，本文提出利用广义分类理论研究紧度量空间上极小同胚间的一种新的渐近共轭关系，并最终利用k -理论数据对这种关系进行刻画。

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.