Operator Algebras and Noncommutative Topology

Blackadar将研究各种类型的算子代数的结构及其与非交换拓扑的关系的一系列问题。主题将包括有限维C*-代数的广义归纳极限和核C*-代数的结构，半射射性，非稳定K-理论和泛系数定理，以及关于C*-代数及其稠密子代数的双变量上同调理论。最近几年彻底改变了算子代数的研究，并在数学和数学物理中产生了一些壮观的应用，将算子代数视为拓扑空间的推广，或者更确切地说，是拓扑空间上连续函数的集合；主要的新特征是，“函数”的乘法不再假定为可变的。人们早就认识到，算符代数为量子力学的数学表述提供了“正确”的框架，最近人们越来越认识到，非对易的“函数空间”(算符代数)自然地出现在各种学科的问题中，如纽结理论(应用于DNA和其他分子结构，以及数学物理)、动力学系统，甚至数理逻辑。显然，有可能对大类中的所有算符代数给出明确的描述，包括应用中出现的大多数代数，这远远超出了几年前人们所认为的可能。这个项目涉及到一个特别重要的类--核C*-代数的分类中的一些中心剩余问题。理解可能发生的数学结构将使我们更深入地了解相关应用问题的性质和行为。

ABSTRACTBlackadarBlackadar will study a range of questions concerning the structure of variousclasses of operator algebras and their relation to noncommutative topology.Topics will include generalized inductive limits of finite-dimensionalC*-algebras and the structure of nuclear C*-algebras, semiprojectivity,nonstable K-theory and the Universal Coefficient Theorem, and bivariantcohomology theories on C*-algebras and their dense subalgebras.An idea which has revolutionized the study of operator algebras in recentyears, and which has led to some spectacular applications throughoutmathematics and mathematical physics, is to view operator algebras asgeneralizations of topological spaces, or more precisely the set ofcontinuous functions on a topological space; the principal new featureis that the multiplication of "functions" is no longer assumed to becommutative. It has long been recognized that operator algebras provide the "right" framework for the mathematical formulation of quantum mechanics, and it has been increasingly recognizedrecently that noncommutative "function spaces" (operator algebras)arise naturally in problems in subjects as diverse as knot theory(with applications to the structure of DNA and other molecules, as wellas mathematical physics), dynamical systems, and even mathematical logic.It has become apparent that it will be possible to give an explicitdescription of all operator algebras in large classes, including mostalgebras arising in applications, far beyond what was thought possibleonly a few years ago. This project concerns some of the centralremaining problems in the classification of one particularly importantclass, the nuclear C*-algebras. Understanding the mathematical structuresthat can occur will give great insight into the nature and behavior ofthe associated applied problems.