Classification of C*-Algebras, Extensions and Homomorphisms

C*-代数的分类、扩展和同态

• 批准号: 9531776
• 负责人: Huaxin Lin
• 金额: $ 3.9万
• 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: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531776&HistoricalAwards=false
• 关键词: Classification; Algebras; Extensions; Homomorphisms

9531776 Lin This project consists of three closely related topics, classification of homomorphisms from one given C*-algebra to another, classification of extensions of one C*-algebra by another, and classification of C*-algebras of real rank zero. The investigator will use KK-theory plus tracial information to determine homomorphisms from an abelian C*-algebra to a simple C*-algebra of real rank zero with stable rank one. To classify extensions, he will classify monomorphisms from an (abelian) C*-algebra to the corona algebra. With additional efforts, by classifying homomorphisms (and automorphisms), one might also be able to classify certain types of C*-algebras (of real rank zero). One particular class that he will consider is direct limits of some extensions of certain algebras. The first example of a C*-algebra is the complex numbers. It is proved to be the only C*-algebra that is also a field. In fact, any finite number of copies of the complex numbers is a C*-algebra. One can also have infinitely many copies. If one glues together continuously an infinite number of copies, one may arrive at the complex-valued continuous functions on an interval, or even complex-valued continuous functions defined on a circle. In fact, every commutative C*-algebra (one in which AB=BA) turns out to be the set of continuous complex-valued functions on some space. By contrast, matrices over the complex field provide noncommutative C*-algebras. Every C*-algebra is technically a normed-closed and conjugate-closed subalgebra of all bounded linear operators on a Hilbert space. What this means is that C*-algebras may be viewed as some kind of generalized complex numbers, and, as in the case of complex numbers, C*-algebras have many important applications, ranging from dynamical systems and quantum mechanics to other fields of mathematics such as operator theory, linear algebra, noncommutative geometry, group representations, and so on. For example, C*- algebras together with their groups of symmetries are often related to problems in dynamical systems. Recent results in C*-algebra theory have been used to answer questions such as when two matrices commute. This project is to study, in a way, how many types of such algebras exist, how to distinguish them from one another, how to construct new C*-algebras from old ones, and to study relationships between these C*-algebras and possible applications to other fields. The objective is to gain a better understanding of these algebras and to develop better theory and methods for applications. ***

9531776 Lin 该项目由三个密切相关的主题组成，从一个给定的 C* 代数到另一个的同态分类，一个 C* 代数由另一个 C* 代数的扩展的分类，以及实秩零的 C* 代数的分类。 研究者将使用 KK 理论加上跟踪信息来确定从阿贝尔 C* 代数到实零阶且稳定一阶的简单 C* 代数的同态。 为了对扩展进行分类，他将对单态从（阿贝尔）C* 代数到电晕代数进行分类。 通过额外的努力，通过对同态（和自同构）进行分类，我们还可以对某些类型的 C* 代数（实秩为零）进行分类。 他将考虑的一个特定类别是某些代数的某些扩展的直接限制。 C* 代数的第一个例子是复数。 它被证明是唯一也是域的 C* 代数。 事实上，任何有限数量的复数副本都是 C* 代数。 一个人还可以拥有无​​限多个副本。 如果将无限数量的副本连续地粘合在一起，则可以得到区间上的复值连续函数，甚至可以得到圆上定义的复值连续函数。 事实上，每个可交换的 C* 代数（其中 AB=BA）都是某个空间上的连续复值函数的集合。 相比之下，复数域上的矩阵提供非交换 C* 代数。 从技术上讲，每个 C* 代数都是希尔伯特空间上所有有界线性算子的范闭和共轭闭子代数。 这意味着 C* 代数可以被视为某种广义复数，并且与复数一样，C* 代数有许多重要的应用，从动力系统和量子力学到其他数学领域，如算子理论、线性代数、非交换几何、群表示等。 例如，C*-代数及其对称群通常与动力系统中的问题相关。 C* 代数理论的最新结果已用于回答诸如两个矩阵何时可交换等问题。 该项目的目的是在某种程度上研究此类代数存在多少种类型、如何区分它们、如何从旧代数构造新的 C* 代数，以及研究这些 C* 代数之间的关系以及在其他领域的可能应用。 目的是更好地理解这些代数并开发更好的应用理论和方法。 ***