C*-algebra theory, Classification and its applications

本项目主要研究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的系统，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，例如动力系统，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，通常存在于无穷多个维度中。“分类”所做的是允许人们从相对少量的相当简单的数值数据中学习C*-代数的身份和结构，类似于指纹用于识别个人的方式。在应用中，C*-代数出现在许多领域，并且经常以伪装的形式出现，因此利用有限的数值数据确定其结构具有很大的优势。本项目的目的是分类迄今为止考虑的最大类的C*-代数，包括代数是特别重要的在许多应用中，这个项目的主要目标是获得一个广泛的分类结果顺从的C*-代数。该策略的一部分是创建新的技术工具，以促进C*-代数及其不变量之间的相互作用。这些工具之一是基本同伦引理的增强，另一个是渐近酉等价的K-理论表征。 另一个重要的目标是验证某些常见的C*-代数实际上确实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。值得注意的是它的一个应用，即非交换同伦理论和动力系统的研究。主要研究者将使用这个广泛的分类理论来研究一个新的关系之间的极小同胚紧度量空间，一个关系称为渐近共轭，并通过K-理论数据来表征它。

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.