Topological and Algebraic Regularity Properties of Nuclear C*-Algebras

数学研究的一般原则是通过对具有高度数学兴趣的对象进行分类来研究它们，也就是说，将每个对象与另一个对象相关联，不变量具有更简单的性质，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。这种方法特别令人满意的是，如果可以从许多角度来看待对象及其不变量，这些角度乍一看非常不同，但却密切相关(有时是出于最微妙和令人惊讶的原因)。我们感兴趣的对象是C*-代数，它允许在拓扑学、群论、分析和动力系统等广泛领域之间建立联系。既然人们不能期望对所有的C*-代数都有一个非平凡的分类不变量，那么基本的问题是：1.什么是可分类的C*-代数的合理类？2.什么是合适的不变量？用K-理论不变量对核C*-代数进行分类，通常被称为‘Elliott程序’，近年来得到了迅速的发展。一方面，核C*-代数的更大的子类被用它们的Elliott不变量来分类；另一方面，这些结果与动力系统理论或Baum-Connes猜想等其他领域进行了卓有成效的联系。此外，分类程序一直是C*-代数理论本身核心的重要新见解的源泉。这项拟议研究的科学目标有三个：a.这一次，我们计划扩展已知的分类结果，并消除一些技术和不必要的限制。最近的PI结果为如何朝着这个方向前进提出了具体的战略。沿途的问题在技术上将是困难的；我们希望汇集该领域领先研究人员的专业知识，如H.Lin、E.Kirchberg、N.C.Phillips、M.Dadarlat和M.Rordam。此外，我们计划建立现有的(和即将到来的)分类结果的新应用，使它们可用于更多来自动力系统以及图论的例子。(对于这项任务，消除如上所述的某些技术限制尤为重要。)我们打算聘请一名博士后研究员，在某一领域具有一些专业知识，以便沿着这些路线应用分类程序。c.最后，我们的目标是统一处理该理论的几个分散部分，这些部分明显相关，但到目前为止还不能以概念性的方式同时处理。我们相信，非对易拓扑维和D-稳定性的新概念能够以令人满意的方式联系在一起，从而使这种统一成为可能。PI与A.Tom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，相关的概念称为“Z-稳定性”、“分解秩数”和“几乎未穿孔的Cuntz半群”。我们希望大幅推广这些成果。与J.Zacharias一起，我计划通过所谓的弱分解秩来建立纯有限和纯无限C*-代数之间的联系。后一个项目还处于起步阶段，但有很高的潜力可以在未来应用。它还为博士项目提供了一些可能的起点。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(如格拉斯哥、阿伯丁或贝尔法斯特)也有强大的小组，但近年来英国研究小组只是略微推动了分类计划的发展。随着PI和J.Zacharias都总部设在诺丁汉，我们希望在分类计划方面建立一个国际领先的中心。

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.