Mathematical Sciences: Operator Algebras and Noncommutative Topology

数学科学：算子代数和非交换拓扑

• 批准号: 9706982
• 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: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706982&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Operator; Algebras; Noncommutative

Abstract Blackadar The investigators propose to continue work on a range of problems concerning the structure of operator algebras and noncommutative algebraic topology. Specific topics to be studied are: structure of inductive limits of C*-algebras, structure of groupoids and their C*-algebras, nonstable K-theory and the fundamental comparability question, and existence and properties of cohomology theories on C*-algebras. If one takes the point of view that a space may be studied via the commutative C*-algebra of complex-valued continuous functions on the space, then the study of noncommutative C*-algebras is a natural generalization. Furthermore, there are numerous situations in ordinary geometry and topology where the natural object of study is a "singular space" which often cannot be studied directly in purely topological terms, but where there is a noncommutative C*-algebra which plays the exact role of the algebra of functions, and where C*-algebra theory has been a crucial tool in advancing the understanding of the topological situation. There has been extensive work in recent years in generalizing notions of algebraic topology, and more recently related notions of differential geometry, to C*-algebras. This work has already paid off spectacularly with fundamental advances in the understanding of the structure of C*-algebras, as well as the beautiful and deep applications in geometry, topology, and mathematical physics. Perhaps the most important result, though, is simply the bridge which has been constructed between operator algebras and topology, and the broad new level of understanding of the interconnections. Virtually all of the research the proposers have done in recent years fits into this general picture of noncommutative topology. The specific projects proposed are continuations or natural outgrowths of previous work, taking into account the way the subject as a whole has developed.

摘要Blackadar研究人员建议继续处理有关操作员代数和非共同代数拓扑结构的一系列问题。 要研究的具体主题是：C* - 代数的归纳限制的结构，群体的结构及其C* - 代数，不稳定的K理论以及基本可比性问题，以及共同体理论在C*-Algebras上的存在和特性。 如果人们认为可以通过在空间上复杂值连续功能的交换性C*代数来研究一个空间，那么对非交通性C* - 代数的研究是自然的概括。 此外，普通几何和拓扑中有许多情况，其中自然研究的对象是一个“奇异空间”，通常无法直接以纯粹的拓扑结构进行研究，但是在函数代数，以及C*-Algebra理论的情况下，有一个不合时宜的C*-Algebra，它扮演着函数代数的确切作用。 近年来，在代数拓扑的概念以及最近相关的差异几何学概念中，已经进行了广泛的工作，以c* - 代数为代数。 这项工作在理解C*代数的结构以及在几何，拓扑和数学物理学中的美丽而深刻的应用方面的理解方面取得了根深蒂固的成就。 不过，也许最重要的结果仅仅是在操作员代数和拓扑之间建造的桥梁，以及对互连的广泛理解水平。 实际上，提议者近年来所做的所有研究都符合这种非交流性拓扑的一般情况。 提议的特定项目是以前工作的连续性或自然产物，考虑到主题的整体发展方式。