Mathematical Sciences: Operator Algebras and Noncommutative Topology

Blackadar的研究人员建议继续研究一系列关于算子代数结构和非对易代数拓扑的问题。具体的研究课题有：C~*-代数的归纳极限的结构，群胚及其C~*-代数的结构，非稳定K-理论和基本比较问题，以及C~*-代数上上同调理论的存在性和性质。如果人们认为一个空间可以通过空间上的复值连续函数的交换C*-代数来研究，那么非交换C*-代数的研究是一个自然的推广。此外，在普通几何和拓扑学中，有许多情况下，研究的自然对象是“奇异空间”，它通常不能直接用纯粹的拓扑术语来研究，但有一个非交换的C*-代数，它扮演着函数代数的确切角色，并且C*-代数理论已经成为促进对拓扑情况的理解的重要工具。近年来，在将代数拓扑的概念和最近相关的微分几何概念推广到C*-代数方面，已经做了大量的工作。这项工作已经在理解C*-代数的结构以及在几何、拓扑和数学物理中美丽而深入的应用方面取得了巨大的进展。然而，也许最重要的结果仅仅是在算子代数和拓扑学之间建立了桥梁，以及对相互联系的广泛的新的理解水平。几乎所有这些提出者最近几年所做的研究都符合非对易拓扑学的一般图景。建议的具体项目是以前工作的延续或自然发展，考虑到整个主题的发展方式。

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.