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

基本信息

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

项目摘要

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.

该项目有两个主要调查线。首先是有限基团对基尔奇伯格代数的作用的分类（纯粹，简单，可分离，核C* - 代数）。与满足通用系数定理的Kirchberg代数的分类非常相似的定理，至少在组是元素循环时，应至少有可能，该动作是在合适的引导类别中（如代数的分类所需的），并且该动作的作用是分离的。人们期望分类为共轭，而不仅仅是共轭。该预期结果的各种扩展也正在研究中。研究的第二大道是对整数组的自由最小作用的转换组C*代数的研究，以及整数组的几个副本在紧凑的度量空间上的乘积。在所有情况下，目的是描述这些代数的结构。对于作用于无限尺寸空间的整数，主要研究者试图将动作的平均维度与交叉产品比较半径联系起来。对于在紧凑的歧管上（以及在其他情况下）的整数副本的乘积，主要研究人员希望证明转换组C*-Algebra在与Jiang-Su代数的张紧下保持稳定，并且具有相关的良好行为。对于同一组，但现在在Cantor套装上行动，主要研究人员想证明，转型组C*-Algebra具有奇特等级的零。该项目将产生更广泛的影响。首先，提议的结果之一将为数学的另一部分提供一个新的链接。其次，首席研究者的先前工作与对电子中电子的量子机械行为的研究有联系，并且该项目中的某些工作有可能进一步阐明该问题。第三，首席调查员积极地在西北太平洋地区为数不多的博士学位的少数机构之一监督研究生。程序。此外，他正在为另外两所大学的学生提供非正式的共同访问。第四，通过与前研究生的持续研究互动，主要研究人员希望帮助支持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能会成为主要研究员大学的本科高级论文。这样的项目使大学生接受了数学研究，从而提高了美国的整体技术准备