Perturbations of Operator Algebras and Related Topics

算子代数的扰动及相关主题

• 批准号: 1101403
• 负责人: Roger Smith
• 金额: $ 19.8万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101403&HistoricalAwards=false
• 关键词: Perturbations; Operator; Algebras; Related; Topics; Perturbations; Operator; Algebras; Related; Topics

This project has two main areas of focus. The first of these concerns the perturbation theory of operator algebras with particular reference to the longstanding Kadison-Kastler problem, which asks whether operator algebras that are close in a suitable metric must be isomorphic (or even unitarily equivalent). This is part of a more general and very difficult problem: when two algebras are the same (isomorphic), how can we recognize that this is so? The difficulty arises because the same algebra can have many different, seemingly unrelated, representations. In recent joint work with coauthors, the principal investigator has made considerable progress on the Kadison-Kastler problem, and he plans to continue this work in several directions: an isomorphism result for close crossed product algebras, the K-theory of close algebras, the shared properties of close algebras, and the question of whether close containments lead to embeddings. Recent progress has come for algebras that have the similarity property, and this has a reformulation in cohomological terms. Thus the second main area of focus will be the Kadison-Ringrose problem, which asks whether certain cohomology groups (which act as obstructions to desirable properties) must vanish. This will involve the theory of completely bounded linear and multilinear maps and also connects to the bounded projection property.The modern study of operator algebras has evolved from two main sources. Matrices, which are generalizations of numbers, were introduced to solve equations and now find applications from computer graphics to search engines for the internet. In formulating quantum mechanics mathematically, von Neumann found that he needed infinite-dimensional versions of matrices called linear operators, which were best studied in operator algebras. Moreover, the time-evolution of quantum mechanical systems came to be expressed in terms of the crossed product by groups of automorphisms. The emerging theory of quantum computation is substantially based on the theory of completely bounded and completely positive maps at the matrix level, and these topics underlie much of the work that will be undertaken. Thus, the results obtained in these areas are likely to impact some of these more concrete areas, since the finite factors are those operator algebras that most closely model matrix algebras. An important aspect of the principal investigator's work has been the training of postdoctoral researchers and doctoral students. Most of the young people who have been mentored by the principal investigator are now in faculty positions where they are training the next generation of scientists and engineers. A scientifically and mathematically trained workforce is essential for the technological future of the country, so the principal investigator will continue to give a central role to the mentoring of young mathematicians.

该项目有两个主要重点领域。其中的第一个涉及运算符代数的扰动理论，特别是涉及长期存在的卡迪森·卡斯特勒问题，该问题询问合适度量中接近的操作员代数是否必须是同构（甚至是单位等效的）。这是一个更普遍和非常困难的问题的一部分：当两个代数相同（同构）时，我们怎么能认识到这是这样？之所以出现困难，是因为相同的代数可以具有许多不同的，看似无关的表示形式。在最近与合着者的联合合作中，首席研究人员在卡迪森 - 卡斯特勒问题上取得了长足的进步，他计划继续朝多个方向继续这项工作：近距离越过代数的同构结果，近代代数的K理论，近代代数的共享特性，近代代数的共享特性，以及近距离包含的问题是否会导致嵌入量的嵌入。具有相似性特性的代数的最新进展取得了进步，这在同一个学角度进行了重新重新制定。因此，重点的第二个主要领域将是kadison-rose问题，该问题询问某些共同体学组（对理想特性的障碍）是否必须消失。这将涉及完全有限的线性和多线性图的理论，并连接到有限的投影属性。对操作员代数的现代研究已从两个主要来源发展。引入了数字概括的矩阵来求解方程，现在从计算机图形技术到搜索引擎搜索Internet的应用程序。在数学上制定量子力学时，冯·诺伊曼（Von Neumann）发现，他需要无限二维版本的矩阵，称为线性算子，这些矩阵是在操作员代数中最好研究的。此外，量子机械系统的时间进化是由自动形态组以交叉产品表示的。量子计算的新兴理论基本上是基于矩阵级别完全有界和完全积极地图的理论，这些主题是将要进行的许多工作的基础。因此，在这些区域获得的结果可能会影响其中一些更具体的区域，因为有限因素是那些最紧密地模拟矩阵代数的操作员代数。主要研究者工作的一个重要方面是对博士后研究人员和博士生的培训。主要研究人员指导的大多数年轻人现在都在教师职位，他们正在培训下一代科学家和工程师。经过科学和数学培训的劳动力对于该国的技术未来至关重要，因此，首席研究人员将继续为年轻数学家的指导发挥核心作用。