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The Cuntz Semigroup and the Fine Structure of Nuclear C*-Algebras

Cuntz 半群和核 C* 代数的精细结构

基本信息

批准号：
EP/I019227/2
负责人：
Joachim Zacharias
金额：
$ 38.5万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI019227%2F2
关键词：
Cuntz Semigroup Fine Structure Nuclear

项目摘要

C*-algebras are norm-closed self-adjoint algebras of operators on Hilbert space. While these are fascinating and richly structured objects themselves, they also provide a natural framework to study connections between such widespread areas as functional analysis, algebra, topology, geometry, geometric group theory, and dynamical systems. Among all C*-algebras, nuclear ones are particularly well-behaved; they can be characterized in many ways, and are accessible to an abundance of techniques, often inspired by (algebraic) topology and geometry. A long-term project in the field is to classify nuclear C*-algebras by K-theoretic data. This is commonly referred to as Elliott programme; it is partially inspired by Connes' celebrated classification of injective factors in the 70s. The programme has seen tremendous progress in past decades, with a particular acceleration in the last 5 years. We now know that classifiability is related to dimension type properties, to tensorial absorption of strongly self-absorbing C*-algebras and to regularity properties of the classifying invariants. We also know that there are examples which cannot be distinguished by traditional K-theoretic invariants. Moreover, the current classification theory works best in the case of simple C*-algebras with an abundance of projections, and the technical difficulties in the non-simple case with few projections are substantial. There is growing body of evidence that a much finer invariant, the Cuntz semigroup, will be crucial to understand the fine structure of nuclear C*-algebras, and ultimately complete the classification problem. In this project we will systematically use Cuntz semigroup techniques to make progress on a range of ambitious problems in the classification programme. More specifically, the scientific aims are threefold. The first two parts below are of a fundamental and groundbreaking nature; the third part aims at applications and concrete classification results:(A) One of the main open problems in the area is to find range results for the Cuntz semigroup, i.e., determine which ordered abelian semigroups can occur as Cuntz semigroups of C*-algebras. The question seems to be extremely hard in general, but range results are indispensable for any successful classification theory, and the Elliott programme is no exception. (B) Many of the currently available classification results for nuclear C*-algebras follow a common pattern: an isomorphism of invariants is lifted to an invertible element of a bivariant theory using the Universal Coefficient Theorem (UCT); the result is then lifted to an isomorphism at the level of algebras. While by now it is clear that the Cuntz semigroup will play an important role as the classifying invariant in future classification results, there still is no bivariant version of it. We plan to develop such a bivariant Cuntz semigroup. We hope that this approach will also shed new light on the behaviour of the Cuntz semigroup with respect to small perturbations, and on the relations between the Cuntz semigroup and nuclearity. (C) In this part of the project we will focus on applications to concrete examples, and on the development of new classification theorems. In particular, we will compute the (bivariant) Cuntz semigroup for new classes of C*-algebras, e.g. for crossed products, for certain non-simple inductive limit C*-algebras, and for non-simple infinite C*-algebras; these results should also spur classification theorems for the same classes of C*-algebras. We will apply Cuntz semigroup techniques to study the fine structure of strongly self-absorbing C*-algebras. One of our motivations here is to make progress on the question whether the known strongly self-absorbing examples really are the only ones; this is related to one of the most important problems in the field, namely whether all nuclear C*-algebras satisfy the UCT.

C*-代数是Hilbert空间上算子的范数闭自共轭代数。虽然这些对象本身是迷人的和丰富的结构对象，但它们也提供了一个自然的框架来研究泛函分析、代数、拓扑、几何、几何群论和动力系统等广泛领域之间的联系。在所有的C*-代数中，核的C*-代数表现得特别好；它们可以用许多方法来刻画，并且可以使用大量的技术，通常是受到(代数)拓扑和几何的启发。该领域的一个长期项目是利用K-理论数据对核C*-代数进行分类。这通常被称为埃利奥特方案；它的部分灵感来自于康尼斯在70年代对内射因素的著名分类。该方案在过去几十年中取得了巨大进展，特别是在过去5年中加速。我们现在知道可分类性与维型性质、强自吸收C*-代数的张量吸收以及分类不变量的正则性有关。我们还知道，有一些例子不能用传统的K-理论不变量来区分。此外，目前的分类理论在具有丰富投影的简单C*-代数的情况下效果最好，而在非简单的情况下具有很少的投影的技术困难是相当大的。越来越多的证据表明，一个更精细的不变量，Cuntz半群，对于理解核C*-代数的精细结构，并最终完成分类问题将是至关重要的。在这个项目中，我们将系统地使用Cuntz半群技术来在分类方案中的一系列雄心勃勃的问题上取得进展。更具体地说，科学目标有三个方面。前两部分是基本的和开创性的；第三部分针对应用和具体的分类结果：(A)该领域的主要公开问题之一是寻找Cuntz半群的值域结果，即确定哪些序交换半群可以作为C*-代数的Cuntz半群出现。总的来说，这个问题似乎非常难，但对于任何成功的分类理论来说，范围结果都是不可或缺的，埃利奥特程序也不例外。(B)许多现有的核C*-代数的分类结果遵循一个共同的模式：利用泛系数定理(UCT)将不变量的同构提升到双变量理论的可逆元；然后将结果提升到代数级别的同构。虽然到目前为止，Cuntz半群作为分类不变量在未来的分类结果中将扮演重要的角色，但仍然没有它的双变量版本。我们计划开发这样一个双变量Cuntz半群。我们希望这一方法也将揭示Cuntz半群在小扰动方面的行为，以及Cuntz半群与核性之间的关系。(C)在本项目的这一部分中，我们将侧重于具体实例的应用，以及新分类定理的发展。特别地，我们将计算新的C*-代数类的(双变量)Cuntz半群，例如对于交叉积，对于某些非简单归纳极限C*-代数，对于非单无限C*-代数，这些结果也应该激励相同类C*-代数的分类定理。我们将应用Cuntz半群技巧来研究强自吸收C*-代数的精细结构。我们在这里的动机之一是要在已知的强自吸收例子是否真的是唯一的问题上取得进展；这涉及到该领域最重要的问题之一，即是否所有的核C*-代数都满足UCT。