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/I019227/1
- 负责人:
- 金额:$ 45.22万
- 依托单位:
- 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:
- 关键词:
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伴代数。虽然这些都是迷人的和丰富的结构化的对象本身,他们也提供了一个自然的框架,研究之间的联系等广泛领域的功能分析,代数,拓扑,几何,几何群论,动力系统。在所有的C*-代数中,核C *-代数表现得特别好;它们可以用许多方式来刻画,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。核C ~*-代数的分类是该领域一个长期的研究课题。这通常被称为艾略特纲领;它部分受到了康纳斯在70年代对内射因子的著名分类的启发。该方案在过去几十年中取得了巨大进展,特别是在过去五年中加速发展。我们现在知道可分类性与维数类型性质、强自吸收C*-代数的张量吸收和分类不变量的正则性有关。我们还知道,有一些例子不能区分传统的K理论不变量。此外,目前的分类理论在具有丰富投影的简单C*-代数的情况下工作得最好,而在具有少量投影的非简单情况下的技术困难是巨大的。有越来越多的证据表明,一个更精细的不变量,Cuntz半群,将是至关重要的理解核C*-代数的精细结构,并最终完成分类问题。在这个项目中,我们将系统地使用Cuntz半群技术,在分类计划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。下面的前两部分是基础性和开创性的;第三部分旨在应用和具体的分类结果:(A)该领域的主要开放问题之一是找到Cuntz半群的值域结果,即,确定C*-代数的哪些序交换半群可以出现为Cuntz半群。这个问题一般说来似乎极其困难,但范围结果对于任何成功的分类理论都是必不可少的,埃利奥特纲领也不例外。(B)许多目前可用的核C*-代数的分类结果遵循一个共同的模式:使用泛系数定理(UCT)将不变量的同构提升为双变理论的可逆元素;然后将结果提升为代数水平上的同构。虽然Cuntz半群作为分类不变量在未来的分类结果中扮演着重要的角色,但至今还没有一个双变的Cuntz半群,我们计划发展这样的双变Cuntz半群.我们希望,这种方法也将揭示新的光的行为的Cuntz半群关于小扰动,以及Cuntz半群和核性之间的关系。(C)在这个项目的这一部分,我们将集中在具体例子的应用,并在新的分类定理的发展。特别地,我们将计算新的C*-代数类的(双变)Cuntz半群,例如交叉积,某些非单归纳极限C*-代数,以及非单无限C*-代数;这些结果也应该刺激相同类别的C*-代数的分类定理。我们将应用Cuntz半群技术来研究强自吸收C*-代数的精细结构。我们在这里的动机之一是在已知的强自吸收的例子是否真的是唯一的问题上取得进展;这与该领域最重要的问题之一有关,即是否所有的核C*-代数都满足UCT。
项目成果
期刊论文数量(10)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
The Cuntz semigroup and stability of close C * -algebras
Cuntz半群和闭C * -代数的稳定性
- DOI:10.2140/apde.2014.7.929
- 发表时间:2014
- 期刊:
- 影响因子:2.2
- 作者:Perera F
- 通讯作者:Perera F
C*-algebras Nearly Contained in Type I Algebras
几乎包含在 I 型代数中的 C* 代数
- DOI:10.4153/cjm-2012-001-1
- 发表时间:2018
- 期刊:
- 影响因子:0
- 作者:Christensen E
- 通讯作者:Christensen E
Rokhlin Dimension and C*-Dynamics
- DOI:10.1007/s00220-014-2264-x
- 发表时间:2015-02
- 期刊:
- 影响因子:2.4
- 作者:Ilan Hirshberg;W. Winter;J. Zacharias
- 通讯作者:Ilan Hirshberg;W. Winter;J. Zacharias
Type II1 factors satisfying the spatial isomorphism conjecture.
II1型因子满足空间同构猜想。
- DOI:10.1073/pnas.1217792109
- 发表时间:2012
- 期刊:
- 影响因子:11.1
- 作者:Cameron J
- 通讯作者:Cameron J
UHF-slicing and classification of nuclear C*-algebras
- DOI:10.1142/s1793525314500198
- 发表时间:2013-07
- 期刊:
- 影响因子:0.8
- 作者:Karen R. Strung;W. Winter
- 通讯作者:Karen R. Strung;W. Winter
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Wilhelm Winter其他文献
Nuclear dimension and $$\mathcal Z$$ -stability
- DOI:
10.1007/s00222-015-0580-1 - 发表时间:
2015-02-27 - 期刊:
- 影响因子:3.600
- 作者:
Yasuhiko Sato;Stuart White;Wilhelm Winter - 通讯作者:
Wilhelm Winter
Designing hybrid AC/DC transmission structures for large electricity systems
- DOI:
10.1007/s00502-020-00837-2 - 发表时间:
2020-10-20 - 期刊:
- 影响因子:0.400
- 作者:
Bernd Klöckl;Georg Deiml;Cora Petino;Wilhelm Winter - 通讯作者:
Wilhelm Winter
Wilhelm Winter的其他文献
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{{ truncateString('Wilhelm Winter', 18)}}的其他基金
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 - 财政年份:2011
- 资助金额:
$ 45.22万 - 项目类别:
Research Grant
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/G014019/1 - 财政年份:2009
- 资助金额:
$ 45.22万 - 项目类别:
Research Grant
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