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.

发现C* - 代数的连续场是用Hausdorff原始频谱强调C*代数的天然结构。但更重要的是，它们已成为各种环境中非交通几何形状的有效工具：电子理论，严格的变形量化，Novikov和Baum-Connes猜想，表示理论和索引理论。 我们将通过具有控制复杂性的连续场研究连续场的近似技术。在许多情况下，可以通过同源方法（捆绑理论，参数化的KK理论）来分析近似领域，从而导致对dixmier和Douady在带有紧凑型操作员的田地上的经典作品的深远概括。我们将与吉姆·麦克卢尔（Jim McClure）合作追求这一研究方向。在不同的方向上，我们提出了一种方法，以证明大量的C* - 代数会在江-SU代数方面吸收。这项研究与安德鲁·汤姆斯（Andrew Toms）和克里斯·菲利普斯（Chris Phillips）合作的目的是根据菲利普斯（Phillips）和Q. lin的结果以及冬季和huaxin lin的最新结果，为与光滑最小动力学系统相关的C*代数分类结果。 量化是在将经典力学与量子力学联系起来的过程中。在此过程中，经典可观察物的交换代数变形为量子可观察物的非共同代数。连续领域的理论为研究这些变形的研究提供了一种可能的数学背景。连续领域的许多多功能性来自以下事实：尽管它们是一般拓扑结构的操作员代数捆绑，但它们不一定是本地琐碎的，因此它们允许仅适用于捕获和传播有趣拓扑不变的变形所必需的适当连续性。拟议的项目旨在为研究人员社区的广泛努力做出贡献，以将数学的经典思想扩展到非共同环境。