Simple Amenable C*-algebras and K-theory

简单可行的 C* 代数和 K 理论

基本信息

批准号：
1665183
负责人：
Huaxin Lin
金额：
$ 18万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665183&HistoricalAwards=false
关键词：
Simple Amenable algebras theory

项目摘要

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.

在量子力学或微观物理世界中，可观察到的可观察到的可以由希尔伯特空间上的自伴操作员建模。此类操作员的系统形成了C*-Algebra。在有限的维度情况下，这些操作员只是矩阵。在更一般的情况下，这些操作员可以由无限矩阵或无限维尺寸希尔伯特空间的矩阵表示。在量子力学的数学公式中，一对代表可观察物的自动伴侣操作员通常是非交通的，与海森伯格的不确定性原理相对应。 C* - 代数的研究通常被视为对拓扑空间中连续功能代数的非共同类似物的研究。有许多来自不同科学领域的C* - 代数，C* - 代数的研究具有多种应用。众所周知，与不同应用外观不同的C* - 代数可能相同。另一方面，来自同一来源的C* - 代数看起来相同，但基本不同的结构。出于应用目的以及理论的角度，从外观中确定C* - 代理的结构非常重要。该项目试图使用少量可计算数据来完全确定某些类别中的C* - 代数。换句话说，可以研究可计算数据以了解C* - 代数的结构。该项目是使用与K理论相关的不变式和分类应用程序对c*代数进行分类的研究。该项目的主要部分是搜索对非正约简单正式c*代数的广泛分类。该策略的一部分是创建新的技术工具来处理非交叉性的非紧凑型空间。目的是建立一种新型的存在定理以及渐近单一简单c* - 代数的渐近单一等效性的K理论表征。一个密切相关的主题是验证某些c* - 代数满足通用系数定理。该项目的研究还将提供新的段落，以极大地简化C*-Algebras理论应用于其他各种相关研究领域的方式，特别是对非交互性同义理论和动力学系统的研究。例如，使用这种广泛的分类理论提出，在紧凑的度量空间上研究最小同构的新关系，该关系称为渐近性共轭，并最终通过k理论数据来表征这种关系。