Classification of C*-Algebras, Extensions and Homomorphisms

点击翻译按钮获取中文摘要

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. ***