Classification of C*-Algebras, Extensions and Homomorphisms

本课题包括三个密切相关的主题：给定C*-代数到另一个C*-代数的同态分类，一个C*-代数被另一个C*-代数扩展的分类，以及实秩0的C*-代数的分类。研究者将使用kk理论加迹信息来确定从一个阿贝尔C*-代数到一个稳定秩为1的实秩0的简单C*-代数的同态。为了对扩展进行分类，他将单态从（阿贝尔）C*-代数分类到冕代数。通过额外的努力，通过对同态（和自同态）进行分类，人们也可能能够对某些类型的C*-代数（实秩为0）进行分类。他将考虑的一个特殊的类别是某些代数的某些扩展的直接极限。C*代数的第一个例子是复数。证明了它是唯一也是域的C*-代数。事实上，任何有限数量的复数副本都是C*-代数。一个人也可以有无限多的副本。如果将无限个副本连续地粘合在一起，就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，每个可交换C*代数（其中AB=BA）都是某个空间上连续复值函数的集合。相反，复域上的矩阵提供非交换的C*-代数。每个C*-代数在技术上都是Hilbert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*-代数可以被看作是某种广义的复数，而且，在复数的情况下，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*-代数及其对称群经常与动力系统中的问题有关。C*-代数理论的最新结果已被用于回答诸如两个矩阵何时交换之类的问题。在某种程度上，这个项目是研究存在多少种这样的代数，如何区分它们，如何从旧的C*-代数中构造新的C*-代数，并研究这些C*-代数之间的关系以及可能在其他领域的应用。目的是更好地理解这些代数，并为应用开发更好的理论和方法。＊＊＊

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. ***