Classification of group actions and structure of transformation group C*-algebras

群作用的分类和变换群C*-代数的结构

• 批准号: 1101742
• 负责人: Norman Phillips
• 金额: $ 17.48万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101742&HistoricalAwards=false
• 关键词: Classification; group; actions; structure; transformation

This project has two main lines of investigation. The first is the classification of actions of finite groups on Kirchberg algebras (purely infinite, simple, separable, nuclear C*-algebras). A theorem very similar to the classification of Kirchberg algebras satisfying the universal coefficient theorem should be possible at the very least when the group is cyclic of prime order, the action is in a suitable bootstrap class (as is required for the classification of algebras), and the action is pointwise outer. One expects classification up to conjugacy, not just cocycle conjugacy. Various extensions of this expected result are also under investigation. The second avenue of research is the study of the transformation group C*-algebras of free minimal actions of the group of integers, and the product of several copies of the group of integers, on compact metric spaces. In all cases, the objective is to describe the structure of these algebras. For the integers acting on an infinite dimensional space, the principal investigator seeks to relate the mean dimension of the action to the radius of comparison of the crossed product. For the product of copies of the integers acting smoothly on a compact manifold (and in some other cases), the principal investigator hopes to prove that the transformation group C*-algebra is stable under tensoring with the Jiang-Su algebra and that it has related good behavior. For the same group, but now acting on the Cantor set, the principal investigator would like to prove that the transformation group C*-algebra has tracial rank zero.This project would have a number of broader impacts. First, one of the proposed results would provide a new and striking link with another part of mathematics. Second, previous work of the principal investigator has connections with the study of the quantum mechanical behavior of an electron in a quasicrystal, and some of the work in this project has the potential to shed further light on this problem. Third, the principal investigator is active in supervising graduate students at one of the few institutions in the Pacific Northwest with a substantial Ph.D. program. Moreover, he is serving as an informal coadvisor to students at two other universities. Fourth, through continued research interaction with former graduate students, the principal investigator expects to help support mathematical research at primarily undergraduate institutions and to foster international cooperation with Mexico. Fifth, some elementary questions and numerical experiments related to the project can become undergraduate senior theses at the principal investigator's university. Such projects expose undergraduates to mathematical research, and thus advance the overall technological preparedness of the U.S.

这个项目有两条主线进行调查。第一个是有限群在Kirchberg代数(纯无限、单、可分、核C*-代数)上作用的分类。一个与满足泛系数定理的Kirchberg代数分类非常相似的定理至少在群是素数阶循环群、作用在适当的Bootstrap类中(就像代数分类所要求的那样)、作用是逐点外作用时是可能的。人们期望分类达到共轭，而不仅仅是共周期共轭。这一预期结果的各种延伸也在调查中。第二条研究途径是研究紧度量空间上整数群的自由极小作用的变换群C*-代数，以及整数群的几个副本的乘积。在所有情况下，目标都是描述这些代数的结构。对于作用在无限维空间上的整数，主要研究者试图将作用的平均维度与交叉积的比较半径联系起来。对于在紧流形上光滑作用的整数的副本的乘积(以及在其他一些情况下)，主要研究者希望证明变换群C*-代数在与酱-苏代数张紧下是稳定的，并且具有相关的良好性态。对于同一个群，但现在作用在Cantor集上，主要研究者想要证明变换群C*-代数具有迹秩零。这个项目将有许多更广泛的影响。首先，其中一个拟议的结果将为数学的另一个部分提供一个新的、引人注目的联系。其次，首席研究员之前的工作与研究准晶中电子的量子力学行为有关，本项目中的一些工作有可能进一步阐明这一问题。第三，首席研究员积极指导太平洋西北部为数不多的拥有大量博士项目的机构之一的研究生。此外，他还担任另外两所大学的学生的非正式顾问。第四，通过继续与以前的研究生进行研究互动，首席研究员希望帮助支持主要是本科机构的数学研究，并促进与墨西哥的国际合作。第五，与该项目相关的一些基本问题和数值实验可以成为主要研究人员大学的本科高级论文。这样的项目让本科生接触到数学研究，从而提高了美国的整体技术准备。

项目成果

K-Theory and the Universal Coefficient Theorem for Simple Separable Exact C*-Algebras Not Isomorphic to Their Opposites

K-理论和简单可分精确C*-代数与逆元不同构的通用系数定理

• DOI: 10.1093/imrn/rnac358
• 发表时间: 2023
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Phillips, N Christopher;Viola, Maria Grazia
• 通讯作者: Viola, Maria Grazia