Topological and Algebraic Regularity Properties of Nuclear C*-Algebras

数学研究的一般原则是通过“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联系起来，即“不变量”，它具有更简单的性质，但保留了有关初始对象的大量信息。然后通过研究初始对象的不变量来比较它们。如果可以从多个角度来观察对象及其不变量，那么这种方法特别令人满意，这些角度乍一看非常不同，但却密切相关（有时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。既然我们不能指望所有C*-代数都有一个非平凡的分类不变量，那么基本问题是：1。可分类的C*代数的合理类别是什么？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“埃利奥特计划”，近年来得到了迅速的发展。一方面，越来越大的核C*-代数的子类已经通过它们的艾略特不变量被分类；另一方面，这些结果与动力系统理论或鲍姆-康恩斯猜想等其他领域产生了富有成效的联系。此外，分类程序一直是对C*-代数本身理论核心的重要新见解的持续来源。拟议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，并消除一些技术和不必要的限制。PI最近的一项结果为如何朝着这个方向前进提出了一个具体的战略。这个过程中的问题在技术上是困难的；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.p illips、M.Dadarlat和M.Rordam。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，使它们可用于来自动力系统和图论的更多示例。（对于这项任务，特别重要的是消除上述某些技术限制。）我们打算聘请一名在某一领域具有一定专业知识的博士后研究员，以便按照这些思路应用分类程序。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，非交换拓扑维数和d -稳定性的新概念可以令人满意地联系起来，使这种统一成为可能。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，并涉及到“z稳定性”、“分解秩”和“几乎无孔孔孔兹半群”等概念。我们希望大大扩展这些成果。我和J.Zacharias一起计划通过所谓的弱分解秩建立纯粹有限和纯粹无限C*-代数之间的联系。后一个项目还处于起步阶段，但在未来的应用中具有很高的潜力。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（如格拉斯哥、阿伯丁或贝尔法斯特）也有强大的研究小组，但近年来英国的研究小组对分类计划的发展只起到了很小的推动作用。由于PI和J.Zacharias都设在诺丁汉，我们希望在分类项目中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.