C*-algebra theory, Classification and its applications

C*-代数理论、分类及其应用

• 批准号: 1361431
• 负责人: Huaxin Lin
• 金额: $ 15万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361431&HistoricalAwards=false
• 关键词: algebra; theory; Classification; its; applications

This project is a study of the classification of C*-algebras and the development of its applications. The study of C*-algebras is a study of some special algebraic systems. These algebraic systems are systems of square matrices, possibly of infinite size. C*-algebras have a strong presence in many diverse fields of science and mathematics such as dynamical systems, noncommutative geometry, and quantum mechanics, to name just a few. C*-algebras are very complicated mathematical objects that usually exist in infinitely many dimensions. What "classification" does is to allow one to learn the identity and structure of a C*-algebra from relatively small amounts of fairly simple numerical data, analogous to the way in which fingerprints are used to identify an individual. In applications, C*-algebras arise in many areas and often appear in disguise, so it is of great advantage to use limited numerical data to determine their structure. The present project aims to classify the largest class of C*-algebras considered thus far, including algebras that are particularly important in many applications.The main goal of this project is to obtain a broad classification result for amenable C*-algebras. Part of the strategy is to create new technical tools to facilitate the interaction between the C*-algebra and its invariants. One of these tools is an enhancement of the Basic Homotopy Lemma and another is a K-theoretic characterization of asymptotic unitary equivalence. Another important objective is to verify that certain commonly appearing C*-algebras actually do belong to our classifiable class. This research will therefore greatly simplify the way in which the theory of C*-algebras is applied. It is worth noting one application, namely, to noncommutative homotopy theory and the study of dynamical systems. The principal investigator will use this broad classification theory to study a new relation among minimal homeomorphisms on a compact metric space, a relation known as asymptotic conjugacy, and to characterize it by means of K-theoretical data.

该项目是对C* - 代数的分类及其应用的开发的研究。 C* - 代数的研究是对一些特殊代数系统的研究。这些代数系统是平方矩阵的系统，可能是无限大小的。 C* - 代数在许多不同的科学和数学领域都有很强的存在，例如动态系统，非交通性几何学和量子力学，仅举几例。 C* - 代数是通常存在于无限多个维度中的非常复杂的数学对象。 “分类”的作用是允许一个人从相对较少的相当简单的数值数据中学习C*代数的身份和结构，类似于使用指纹识别个体的方式。在应用程序中，c* - 代数在许多领域都出现，并且经常出现变相，因此使用有限的数值数据来确定其结构是很大的优势。本项目旨在对迄今为止考虑的最大的C* - 代数类别进行分类，其中包括在许多应用中尤为重要的代数。该项目的主要目标是为Amenable C*-ergebras获得广泛的分类结果。策略的一部分是创建新的技术工具，以促进C*-Algebra及其不变性之间的相互作用。这些工具之一是增强基本同型引理的能力，另一个工具是渐近单一等效性的K理论表征。 另一个重要的目标是验证某些通常出现的C*-Algebras实际上确实属于我们的分类类。因此，这项研究将大大简化C* - 代数理论的应用方式。值得注意的是，一种应用于非共同同质理论和动态系统的研究。首席研究者将使用这种广泛的分类理论来研究紧凑型度量空间上最小同构的新关系，即一种称为渐近偶联性的关系，并通过k理论数据来表征它。