Topological and Algebraic Regularity Properties of Nuclear C*-Algebras

核 C* 代数的拓扑和代数正则性质

• 批准号: EP/G014019/2
• 负责人: Wilhelm Winter
• 金额: $ 2.82万
• 依托单位: University of Münster
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2011
• 资助国家: 英国
• 起止时间: 2011 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG014019%2F2
• 关键词: Topological; Algebraic; Regularity; Properties; Nuclear

A general principle of mathematical research is to study objects of high mathematical interest by 'classifying' them, that is, associate to each object another one, the 'invariant', which is of a simpler nature, yet retains a substantial amount of information about the initial object. One then seeks to compare the initial objects by studying their invariants. Such an approach is particularly satisfying if it is possible to view the objects and their invariants from a number of perspectives which are very different at first glance, yet are intimately related (sometimes for most subtle and surprising reasons).Our objects of interest are 'C*-algebras', which allow connections between such widespread areas as topology, group theory, analysis and dynamical systems. Since one cannot hope for a nontrivial classifying invariant for all C*-algebras, the fundamental questions then are: 1. What are reasonable classes of C*-algebras that are classifiable? 2. What are the suitable invariants?The classification of nuclear C*-algebras by K-theoretic invariants, commonly referred to as `Elliott program', has seen rapid development in recent years. On the one hand, ever larger subclasses of nuclear C*-algebras have been classified by their Elliott invariants; on the other hand, these results have made fruitful contact with other areas like the theory of dynamical systems or the Baum-Connes conjecture. Furthermore, the classification program has been a constant source of important new insights into the heart of the theory of C*-algebras itself. The scientific aim of the proposed research is threefold:A. For once, we are planning to extend the known classification results and to remove some technical and unnecessary constraints. A recent result of the PI suggests a concrete strategy of how to proceed in this direction. The problems along the way will be technically hard; we are hoping to bring together the expertise of leading researchers in the field, such as H.Lin, E.Kirchberg, N.C.Phillips, M.Dadarlat and M.Rordam. B. Furthermore, we plan to establish new applications of the existing (and upcoming) classification results to make them available to more examples from dynamical systems as well as graph theory. (For this task, it will be particularly important to remove certain technical constraints as indicated above.) We intend to hire a postdoctoral researcher with some expertise in a field allowing for applications of the classification program along these lines.C. Finally, we are aiming at a unified treatment of several scattered parts of the theory which are clearly related, but so far cannot be handled simultaneously in a conceptual way. We are confident that the new notions of noncommutative topological dimension and of D-stability can be connected in a satisfactory manner to make such a unification possible. The joint results of the PI with A.Toms and with M.Rordam have prepared ground for a conceptual treatment of purely finite C*-algebras, relating concepts called 'Z-stability', 'decomposition rank' and 'almost unperforated Cuntz semigroup'. We hope to substantially extend these results. Together with J.Zacharias I plan to establish a connection between purely finite and purely infinite C*-algebras via the so-called weak decompositon rank. This latter project is at its very beginnings, but has a high potential to allow for future applications. It also offers a number of possible starting points for PhD projects.At a strategic level, the project aims at extending the UK's internationally leading role in the theory of C*-algebras. While there are strong groups in other areas of the field (e.g. in Glasgow, Aberdeen, or Belfast), the development of the classification program in recent years has only marginally been driven by research groups from the UK. With the PI and J.Zacharias both based in Nottingham, we hope to establish an internationally leading centre in the classification program.

数学研究的一个一般原则是通过“分类”来研究具有高度数学兴趣的对象，也就是说，将每个对象与另一个对象相关联，即“不变量”，它具有更简单的性质，但保留了关于初始对象的大量信息。然后，人们试图通过研究它们的不变量来比较初始对象。这种方法是特别令人满意的，如果它是可能的，以查看对象和它们的不变量从许多角度看，这是非常不同的，乍一看，但密切相关的（有时是最微妙的和令人惊讶的原因）。我们感兴趣的对象是'C*-代数'，它允许连接等广泛的领域，如拓扑，群论，分析和动力系统。由于不能期望所有的C*-代数都有一个非平凡的分类不变量，那么基本问题是：1。什么是C*-代数的合理类是可分类的？2.什么是合适的不变量？核C*-代数的K-理论不变量分类，通常被称为“Elliott程序”，近年来得到了迅速的发展。一方面，核C*-代数的更大的子类已经被它们的Elliott不变量分类;另一方面，这些结果与其他领域如动力系统理论或Baum-Connes猜想有着富有成效的联系。此外，分类程序一直是对C*-代数理论本身核心的重要新见解的不断来源。这项研究的科学目的有三个方面：A.这一次，我们计划扩展已知的分类结果，并删除一些技术和不必要的限制。PI最近的一项研究结果提出了一个如何朝着这个方向前进的具体战略。沿着的问题在技术上是困难的;我们希望能汇集该领域主要研究人员的专门知识，如H.林、E.基希贝格、N. C.菲利普斯、M.达达拉特和M.罗丹。B。此外，我们计划建立现有（和即将到来的）分类结果的新应用程序，使它们可用于动力系统以及图论的更多示例。(For在这项任务中，消除上述某些技术限制将特别重要。我们打算聘请一位博士后研究员，他在某一领域有一定的专业知识，可以沿着沿着这些路线应用分类程序。最后，我们的目标是统一处理理论的几个分散的部分，这些部分显然是相关的，但到目前为止还不能以概念的方式同时处理。我们相信，新的概念的非交换拓扑维数和D-稳定性可以连接在一个令人满意的方式，使这样的统一可能。PI与A.Toms和M.Rordam的联合结果为纯有限C*-代数的概念处理奠定了基础，相关概念称为“Z-稳定性”，“分解秩”和“几乎无孔Cuntz半群”。我们希望能大大扩展这些成果。我和J.Zacharias计划通过所谓的弱分解秩在纯有限和纯无限C*-代数之间建立联系。后一个项目才刚刚开始，但有很大的潜力，以允许未来的应用。它还为博士项目提供了许多可能的起点。在战略层面上，该项目旨在扩大英国在C*-代数理论方面的国际领先地位。虽然在该领域的其他领域（例如，在格拉斯哥、阿伯丁或贝尔法斯特）也有强大的团体，但近年来分类计划的发展仅在很小程度上受到英国研究团体的推动。随着PI和J.Zacharias都设在诺丁汉，我们希望在分类计划中建立一个国际领先的中心。