Classification of amenable C*-algebras and applications

本文通过比较两个C*-代数的k理论数据来研究它们同构的情况。特别地，它试图用k理论数据来确定两个近似可分（或z稳定）且满足普适系数定理的单可分简单可调C*-代数是否同构。并提出了一个密切相关的问题，即最小动力系统的k理论数据能否通过与系统的其他k理论数据的关联变换C*-代数来确定最小动力系统的结构。将C*-代数视为非交换拓扑空间，提出研究C*-代数中的（近似）同伦理论。根据狄拉克和冯·诺伊曼的说法，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。这样的系统具有和复数系统一样的加法和乘法结构。然而，在C*-代数中，乘法不一定是可交换的，这与海森堡测不准原理相对应。设X是紧度量空间F是一个从X到X的变换假设它是可逆的F和它的逆都是连续的。对（X， F）形成相关的变换C*-代数。为了研究（X， F）的动力结构，可以从相关的C*-代数开始。对相关C*-代数结构的研究提供了原始动力系统的信息。其中一个例子是X是单位圆，F是圆上的不合理旋转。所关联的C*-代数是一个一元可分的简单可服从C*-代数。这种C*-代数也可以由两个幺正算子的典型非交换关系构成。它也被称为非交换环面。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。例如，C*-代数可以由某些希尔伯特空间上的算子、经典动力系统、非交换几何、群表示或许多其他研究（如量子化）形成。对一类C*-代数进行分类，就是利用少量的可计算数据完全确定该类中的C*-代数及其结构，在此过程中，人们还可以了解希尔伯特空间、动力系统、非交换几何、群表示上的相关算子，进而为科学领域的其他领域提供进一步的应用。

This proposal is to study when two C*-algebras are isomorphic by comparing their K-theoretic data. In particular, it attempts to use the K-theoretic data to determine whether two unital separable simple amenable C*-algebras which are approximately divisible (or Z-stable) and satisfies the Universal Coefficient Theorem are isomorphic. It also proposes to study a closely related problem whether K-theoretic data of a minimal dynamic system could determine the structure of the minimal dynamic system by studying the associated transformation C*-algebra together with other K-theoretic data of the system. Viewing C*-algebras as non-commutative topological spaces, it also proposes to study (approximate) homotopy theory in C*-algebras.In the micro-scopical physical world, an observable may be modeled by a self-adjoint operator on a Hilbert space, according to Dirac and von Neumann. A system of such operators forms a C*-algebra. Such a system has the structure of addition and multiplication like the system of complex numbers. However, in a C*-algebra, multiplication may not be commutative which corresponds to the Heisenberg uncertainty principle. Let X be a compact metric space and F be a transformation from X to X which is assumed to be invertible and both F and its inverse are continuous. The pair (X, F) forms the associated transformation C*-algebra. To study the dynamical structure of (X, F), one may start with the associated C*-algebra. The study of the structure of the associated C*-algebra provides the information of the original dynamical system. One of such examples is the special case that X is the unit circle and F is an irrational rotation on the circle. The associated C*-algebra is a unital separable simple amenable C*-algebra. This C*-algebra can also be formed by a typical non-commutative relation of two unitary operators. It is also known as non-commutative torus. There are many C*-algebras come from different fields of sciences and the study of C*-algebras has variety of applications. For example, C*-algebras may be formed by operators on some Hilbert spaces, by classical dynamic systems, by non-commutative geometry, by group representations, or, by many other studies such as quantization. To classify a class of C*-algebras is to use a few computable data to completely determine C*-algebras in the class and their structure, in the process, one may also understand the related operators on Hilbert spaces, dynamical systems, non-commutative geometry, group representations, and, in turn, these may further provide applications to other parts of the scientific world.