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

C*-代数是Hilbert空间上算子的范闭自伴随代数。虽然这些都是令人着迷且结构丰富的对象，但它们也为研究泛函分析、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提供了一个自然的框架。在所有的C*代数中，核代数表现得特别好；它们可以以多种方式表征，并且可以使用大量的技术，通常受到（代数）拓扑和几何的启发。利用k理论数据对核C*-代数进行分类是该领域的一个长期课题。这通常被称为艾略特方案；部分灵感来自于Connes在70年代著名的注射因子分类。该计划在过去几十年里取得了巨大进展，在过去5年里尤其加速。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。我们也知道有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*-代数情况下效果最好，而在具有较少投影的非简单情况下技术困难很大。越来越多的证据表明，一个更精细的不变量——昆兹半群，对于理解核C*-代数的精细结构，并最终完成分类问题至关重要。在这个项目中，我们将系统地使用Cuntz半群技术来解决分类程序中一系列雄心勃勃的问题。更具体地说，科学目标有三个方面。下面的前两部分是基础性和开创性的；(A)该领域的主要开放问题之一是寻找Cuntz半群的范围结果，即确定哪些有序阿贝尔半群可以作为C*-代数的Cuntz半群出现。一般来说，这个问题似乎极其困难，但范围结果对于任何成功的分类理论都是必不可少的，艾略特程序也不例外。(B)目前许多核C*-代数的分类结果都遵循一个共同的模式：使用通用系数定理（UCT）将不变量的同构提升到双变理论的可逆元；然后将结果提升到代数水平上的同构。虽然现在已经很清楚，在未来的分类结果中，Cuntz半群将作为分类不变量发挥重要作用，但目前还没有它的双变版本。我们计划发展这样一个双变Cuntz半群。我们希望，这一方法也将对孔茨半群在小扰动下的行为以及孔茨半群与核之间的关系提供新的启示。(C)在项目的这一部分中，我们将侧重于具体例子的应用，以及新的分类定理的发展。特别地，我们将计算新一类C*-代数的（双变）Cuntz半群，例如交叉积，某些非简单归纳极限C*-代数，以及非简单无限C*-代数；这些结果也应该刺激C*-代数的分类定理。我们将应用Cuntz半群技术研究强自吸收C*-代数的精细结构。我们在这里的动机之一是在这个问题上取得进展是否已知的强自吸收的例子真的是唯一的；这关系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。

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.