Classifying *-Homomorphisms

对*-同态进行分类

• 批准号: 2000129
• 负责人: Christopher Schafhauser
• 金额: $ 17.86万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000129&HistoricalAwards=false
• 关键词: Classifying; Homomorphisms

This project relates to the classification and structure of amenable operator algebras. Operator algebras is an area of mathematics which began with the work of Murray and von Neumann in the 1930's motivated in part by Heisenberg's approach to quantum mechanics in terms of infinite matrices. Von Neumann's development of operators on a Hilbert space put Heisenberg's ideas on rigorous foundations. In this theory, the observable data is represented by certain operators (i.e. infinite matrices) on a Hilbert space. One of the most famous concepts of quantum mechanics is Heisenberg's uncertainty principle that the speed and position of a particle cannot be known simultaneously. The mathematically rigorous version of this statement is that the operators P and Q which measure position and momentum do not commute; i.e. PQ and QP are not equal (there is, however, a precise formula relating P and Q). Operator algebras is the study of algebraic relations between collections of operators. Amenable operators algebras form a particularly important class of operators algebras. In recent years, there has been a substantial work in classifying simple amenable operator algebras. In a certain sense, the simple examples are the basic building blocks of the theory. The goal of this project is to understand the relations between the simple amenable operator algebras (such as ways one can be embedded into the other) and the symmetries of such algebras and to exploit these ideas to uncover structural information about operator algebras as a whole. This project also will be contributing to the education of the US workforce through the training of graduate students.More technically, recent progress in Elliott's Program shows separable, simple, nuclear, regular C*-algebras in the UCT class are classified up to isomorphism via their operator K-theory groups, their trace simplex, and the pairing between them. This should be viewed as a direct analogue to the Connes-Haagerup classification of separably acting injective factors in terms of their type and flow of weights. The Connes-Haagerup classification has become a corner stone of modern von Neumann algebra theory, and the C*-algebraic analogue may be expected to have an equally important role in C*-algebra theory. In recent joint work with Carrion, Gabe, Tikuisis, and White, the PI showed that embeddings of "classifiable" C*-algebras are determined up to approximate unitary equivalence by K-theoretic data. The aim of this project is to refine and expand on these techniques with an eye towards equivariant classification and non-simple classification, as well as applications to the regularity theory, such as computing the nuclear dimension of a new classes of C*-algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目涉及顺从算子代数的分类和结构。 算子代数是数学的一个领域，开始于默里和冯诺依曼在20世纪30年代的工作，部分动机是海森堡的方法量子力学方面的无限矩阵。 冯诺依曼的发展运营商的希尔伯特空间把海森堡的想法严格的基础。 在该理论中，可观测数据由希尔伯特空间上的某些算子（即无限矩阵）表示。 量子力学中最著名的概念之一是海森堡的不确定性原理，即粒子的速度和位置不能同时知道。 这个陈述的数学严格版本是测量位置和动量的算子P和Q不对易;即PQ和QP不相等（然而，有一个精确的公式将P和Q联系起来）。 算子代数是研究算子集合之间的代数关系的学科。 顺从算子代数是一类特别重要的算子代数. 近年来，在简单顺从算子代数的分类方面做了大量的工作。在某种意义上，简单的例子是理论的基本组成部分。 这个项目的目标是了解简单的顺从算子代数之间的关系（例如一个可以嵌入到另一个中的方式）和这种代数的对称性，并利用这些想法来揭示整个算子代数的结构信息。该项目还将通过培养研究生为美国劳动力教育做出贡献。更技术性的是，Elliott计划的最新进展表明，UCT类中的可分、简单、核、正则C*-代数通过它们的算子K-理论群、迹单形和它们之间的配对被分类为同构。 这应该被看作是一个直接的模拟康纳斯-哈格鲁普分类可分离作用内射因子的类型和流量的权重。 Connes-Haagerup分类已经成为现代冯·诺依曼代数理论的基石，而C*-代数类似物可能会在C*-代数理论中扮演同样重要的角色。 在最近与Carrion，Gabe，Tikuisis和白色的联合工作中，PI表明“可分类”C*-代数的嵌入由K-理论数据确定为近似酉等价。 该项目的目的是完善和扩展这些技术，着眼于等变分类和非简单分类，以及正则性理论的应用，例如计算一类新的C*-代数的核维数。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。