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. ***

小行星9531776 这个项目包括三个密切相关的主题，分类同态从一个给定的C*-代数到另一个，分类的扩展一个C*-代数的另一个，分类的C*-代数的真实的秩零。 研究者将使用KK-理论加上迹信息来确定从交换C*-代数到真实的秩为零且稳定秩为一的简单C*-代数的同态。 为了分类扩展，他将分类单态从（交换）C*-代数的冠代数。 通过额外的努力，通过对同态（和自同构）进行分类，人们也可以对某些类型的C*-代数（真实的秩为零）进行分类。 一个特殊的类，他将考虑直接限制的一些扩展某些代数。 C*-代数的第一个例子是复数。 它被证明是唯一的C*-代数，也是一个域。 事实上，复数的任何有限数量的拷贝都是 C*-代数 一个人也可以有无限多个副本。 如果 一个人连续地将无限多个副本粘在一起，可以得到区间上的复值连续函数，甚至可以得到定义在圆上的复值连续函数。 事实上，每个交换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。 每个C*-代数是Hilbert空间上所有有界线性算子的赋范闭共轭闭子代数。 这意味着C*-代数可以被看作 作为某种广义复数，并且，在这种情况下， 在复数中，C*-代数有许多重要的应用， 从动力系统和量子力学到其他领域 数学，如算子理论，线性代数，非交换 几何、群表示等。例如，C*-代数 和它们的对称群一起， 在动力系统中。 C*-代数理论的最新结果是 用来回答诸如两个矩阵可交换的问题。 这 项目是研究，在某种程度上，有多少类型的这样的代数存在， 如何区分它们，如何构建新的 C*-代数，并研究它们之间的关系 C*-代数及其在其他领域的可能应用。 客观 是为了更好地理解这些代数， 更好的理论和方法的应用。 ***