Operator Algebras and K-theory

C*-代数的连续域被发现为具有Hausdorff本原谱的C*-代数的自然结构。但更重要的是，它们已经成为非对易几何在一系列背景下的有效工具：E理论、严格形变量子化、Novikov猜想和Baum-Connes猜想、表示理论和指数理论。我们将研究连续场与复杂性可控的连续场的逼近技术。在许多情况下，近似场可以用同调方法(鞘理论、参数KK理论)来分析，这导致了Dixmier和Douady关于纤维紧算子场的经典工作的深远推广。我们将与吉姆·麦克卢尔合作，继续这一研究方向。在不同的方向上，我们提出了一种证明大类C*-代数张量吸收酱-苏代数的方法。这项研究的目的是在Phillips和Q.Lin的结果以及温特和林华新的最新结果的基础上，给出与光滑极小动力系统相关的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。在这个过程中，经典可观测的交换代数被变形为量子可观测的非交换代数，连续场理论为研究这些变形提供了一种可能的数学背景。连续场的多功能性很大程度上来自于这样一个事实：虽然它们是一般拓扑学意义上的算子代数的丛，但它们不一定是局部平凡的，因此它们只允许捕捉和传播有趣的拓扑不变量的变形所需的适量的连续性。这项拟议的项目旨在促进研究人员社区的广泛努力，将经典的数学思想推广到非对易环境中。

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.