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这个项目由三个密切相关的主题组成，分类从一个给定的C*-Algebra到另一个给出的同构，另一个c*-algebra的扩展分类，另一个c*-algebra的扩展，以及对真实等级零的C* - 代数的分类。 研究者将使用KK理论加纹理信息来确定从Abelian c*-Algebra到一个简单的C*-CH-Algebra的同态零等级为零，而稳定等级。 为了对扩展进行分类，他将对（Abelian）C*-Algebra到Corona代数进行分类。 通过额外的努力，通过对同态（和自动形态）进行分类，也许还可以对某些类型的C*-Algebras（实际等级零）进行分类。 他将考虑的一个特定类是某些代数的某些扩展的直接限制。 c* - 代数的第一个示例是复数。 事实证明，这也是唯一也是一个领域的C*-Algebra。 实际上，任何有限数量的复数副本都是C*-代数。 一个人也可以无限地拥有许多副本。 如果一个人连续地胶合在一起，则可以在一个间隔上到达复杂值的连续功能，甚至可以在圆上定义的复杂值的连续函数。 实际上，每个换向的C*-Algebra（其中一个AB = BA）都是在某些空间上的连续复杂值函数的集合。 相比之下，复杂场上的矩阵提供了非交通性的C* - 代数。 从技术上讲，每个C* - 代数都是Hilbert空间上所有有界线性算子的标准锁定和共轭封闭的子代数。 这意味着C*-Algebras可能被视为某种普遍的复数数字，并且与复杂数字一样，C*-Algebras具有许多重要的应用，从动力学系统和量子力学到其他数学领域，例如操作员理论等数学领域，线性代数，非公认的几何形式，组成，组代表和等等。 例如，C* - 代数及其对称组通常与动态系统中的问题有关。 C* - 代数理论的最新结果已被用来回答诸如两个矩阵通勤时间之类的问题。 该项目是在某种程度上研究存在多少种类型的代数，如何将它们与彼此区分开，如何构建新的C* - 代数与旧代数，并研究这些C* - 代数之间的关系以及可能对其他领域的应用程序之间的关系。 目的是更好地理解这些代数，并为应用提供更好的理论和方法。 ***