Operator Algebras and K-theory

算子代数和 K 理论

• 批准号: 0801173
• 负责人: Marius Dadarlat
• 金额: $ 29.57万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801173&HistoricalAwards=false
• 关键词: Operator; Algebras; theory

Continuous fields of C*-algebras were discovered as natural structures that underline C*-algebras with Hausdorff primitive spectrum. But more importantly, they have become effective tools of noncommutative geometry in a large array of contexts: E-theory, strict deformation quantization, the Novikov and the Baum-Connes conjectures, representation theory and index theory. We will study approximation techniques of continuous fields by continuous fields with controlled complexity. In many instances, the approximating fields can be analyzed by homological methods (sheaf theory, parametrized KK-theory), leading to far-reaching generalizations of the classical work of Dixmier and Douady on fields with fibres the compact operators. We will pursue this direction of research in collaboration with Jim McClure. In a different direction we propose an approach for proving that large classes of C*-algebras absorb tensorially the Jiang-Su algebra. The goal of this research in collaboration with Andrew Toms and Chris Phillips is to give classification results for C*-algebras associated with smooth minimal dynamical systems, based on results of Phillips and Q. Lin and very recent results of Winter and Huaxin Lin. Quantization arises in the process of relating classic mechanics to quantum mechanics. In this process, the commutative algebra of classical observables is deformed into a noncommutative algebra of quantum observables.The theory of continuous fields provides one possible mathematical context for the study of these deformations. Much of the versatility of continuous fields comes from the fact that while they are bundles of operator algebras in the sense of general topology, they are not necessarily locally trivial and hence they allow for just the right amount of continuity necessary for deformations that capture and propagate interesting topological invariants. The proposed project aims to contribute to the extensive effort of a community of researchers to extend classical ideas of mathematics to noncommutative contexts.

C*-代数的连续场被发现为具有Hausdorff原始谱的C*-代数下的自然结构。但更重要的是，它们已经成为非交换几何在大量背景下的有效工具：e理论，严格变形量化，诺维科夫和Baum-Connes猜想，表示理论和指标理论。我们将研究用控制复杂度的连续场逼近连续场的技术。在许多情况下，近似场可以用同调方法（束理论，参数化kk理论）进行分析，从而对Dixmier和Douady关于紧算子纤维场的经典工作进行了深远的推广。我们将与吉姆·麦克卢尔（Jim McClure）合作，朝着这个方向进行研究。在一个不同的方向上，我们提出了一种方法来证明大类C*-代数是张性地吸收Jiang-Su代数的。这项研究的目标是与Andrew Toms和Chris Phillips合作，基于Phillips和Q. Lin的结果以及Winter和Huaxin Lin最近的结果，给出与光滑最小动力系统相关的C*代数的分类结果。量子化是在将经典力学与量子力学联系起来的过程中产生的。在此过程中，经典可观测量的交换代数被变形为量子可观测量的非交换代数。连续场理论为研究这些变形提供了一个可能的数学背景。连续场的通用性很大程度上来自于这样一个事实，即虽然它们是一般拓扑意义上的算子代数束，但它们不一定是局部平凡的，因此它们允许适当数量的连续性，以实现捕获和传播有趣的拓扑不变量的变形。提议的项目旨在为研究人员社区的广泛努力做出贡献，将经典数学思想扩展到非交换环境。