Operator Algebras and Noncommutative Topology

算子代数和非交换拓扑

• 批准号: 0070763
• 负责人: Bruce Blackadar
• 金额: $ 9万
• 依托单位: Board of Regents, NSHE, obo University of Nevada, Reno
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070763&HistoricalAwards=false
• 关键词: Operator; Algebras; Noncommutative; Topology

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.

Blackadar将研究算子代数的各种类的结构及其与非交换拓扑的关系，包括有限维C *-代数的广义归纳极限和核C*-代数的结构，半投射性，非稳定K-理论和泛系数定理，C ~*-代数及其稠密子代数上的双变上同调理论，这一思想使近年来算子代数的研究发生了革命性的变化，并在数学和数学物理中得到了一些引人注目的应用，就是把算子代数看作拓扑空间的推广，或者更确切地说，看作拓扑空间上的连续函数的集合;主要的新特点是不再假定“函数”的乘法是不可变的。 人们早就认识到，算子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“函数空间”（算子代数）自然地出现在像纽结理论这样不同的问题中（应用于DNA和其他分子的结构，以及数学物理），动力系统，很明显，我们有可能对大类中的所有算子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的想象。 这个项目涉及的一些centralremaining问题的分类一个特别重要的类，核C*-代数。 理解可能出现的数学结构将使我们深入了解相关应用问题的性质和行为。