Classification of amenable C*-algebras and applications

适合的 C* 代数分类和应用

• 批准号: 0754813
• 负责人: Huaxin Lin
• 金额: $ 14.4万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0754813&HistoricalAwards=false
• 关键词: Classification; amenable; algebras; applications

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.

该建议是研究两个C* - 代数何时是同构，通过比较它们的K理论数据。特别是，它试图使用K理论数据来确定两个近似可分开（或Z稳定）并满足通用系数定理的两个Unital可分离的简单的c* - 代数是否同构。它还提出了一个密切相关的问题，是否可以通过研究相关的转换C*-Algebra以及系统的其他K理论数据来确定最小动态系统的k理论数据是否可以确定最小动态系统的结构。 根据Dirac和Voneumann的说法，将C* - 代数视为非共同拓扑空间，它还提议研究（近似）同质理论在C*-Algebras中。此类操作员的系统形成了C*-Algebra。这样的系统具有添加和乘法的结构，例如复数系统。但是，在c* - 代数中，乘法可能不相当，这与海森堡的不确定性原理相对应。令X为紧凑的度量空间，F为从X到X的转换，假定可逆，而F及其逆也是连续的。这对（x，f）形成了相关的转换C*-Algebra。 为了研究（x，f）的动力结构，可以从相关的c* - 代数开始。 相关C*-Algebra结构的研究提供了原始动力学系统的信息。这样的例子之一是X是单位圆，F是圆圈上的不合理旋转。相关的c* - 代数是一个Unital可分离的简单的Amenable c*-ergebra。该c* - 代数也可以通过两个单一操作员的典型非交通关系形成。它也被称为非交流性圆环。有许多C* - 代数来自不同的科学领域，而C* - 代数的研究具有多种应用。例如，c* - 代数可以是由经营者在某些希尔伯特空间，经典动态系统，非交通性几何形状，分组表示或其他许多其他研究（例如量化）中形成的。为了对一类C* - 代数进行分类，是使用一些可计算数据来完全确定类及其结构中的C* - 代数，在此过程中，人们还可以理解希尔伯特空间，动力学系统，非交互性几何学，团体表征的相关操作员，而这些又可能进一步提供有关科学世界的其他部分的应用。