C*-algebra theory, Classification and its applications

本课题是研究C*-代数的分类及其应用的发展。C*代数的研究是对一些特殊代数系统的研究。这些代数系统是方阵系统，可能是无限大的。C*-代数在许多不同的科学和数学领域都有很强的存在，比如动力系统、非交换几何和量子力学，仅举几例。C*代数是非常复杂的数学对象，通常存在于无限多个维度中。“分类”所做的是允许人们从相对少量的相当简单的数字数据中了解C*-代数的身份和结构，类似于使用指纹来识别个人的方式。在应用中，C*-代数出现在很多领域，而且常常以伪装的形式出现，因此利用有限的数值数据来确定其结构具有很大的优势。目前的项目旨在分类迄今为止考虑过的最大的C*-代数类，包括在许多应用中特别重要的代数。本项目的主要目标是获得可修正C*-代数的广泛分类结果。该策略的一部分是创建新的技术工具来促进C*-代数与其不变量之间的交互。这些工具之一是对基本同伦引理的改进，另一个是对渐近酉等价的k理论刻画。另一个重要的目标是验证某些通常出现的C*-代数实际上属于我们的可分类类。因此，这项研究将大大简化C*-代数理论的应用方式。值得注意的是它在非交换同伦理论和动力系统研究中的一个应用。主要研究者将利用这一广义分类理论研究紧度量空间上极小同胚间的一种新关系，即渐近共轭关系，并利用k -理论数据对其进行表征。

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.