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

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。第一个涉及算子代数的摄动理论，特别涉及长期存在的 Kadison-Kastler 问题，该问题询问在合适度量中接近的算子代数是否必须同构（甚至酉等价）。这是一个更普遍且非常困难的问题的一部分：当两个代数相同（同构）时，我们如何认识到事实如此？困难的出现是因为同一个代数可以有许多不同的、看似无关的表示。在最近与合著者的合作中，主要研究者在 Kadison-Kastler 问题上取得了相当大的进展，他计划在几个方向上继续这项工作：闭交积代数的同构结果、闭代数的 K 理论、封闭代数的共同属性，以及封闭包含是否导致嵌入的问题。具有相似性质的代数最近取得了进展，并且用上同调术语进行了重新表述。因此，第二个主要关注领域将是 Kadison-Ringrose 问题，该问题询问某些上同调群（对理想性质的阻碍）是否必须消失。这将涉及完全有界线性和多线性映射的理论，并且还涉及有界投影性质。算子代数的现代研究有两个主要来源。矩阵是数字的概括，最初是为了求解方程而引入的，现在它的应用范围从计算机图形学到互联网搜索引擎。在以数学方式表述量子力学时，冯·诺依曼发现他需要称为线性算子的矩阵的无限维版本，这在算子代数中得到了最好的研究。此外，量子力学系统的时间演化开始用自同构群的叉积来表达。新兴的量子计算理论基本上基于矩阵级别的完全有界和完全正映射的理论，这些主题是即将开展的大部分工作的基础。因此，在这些领域获得的结果可能会影响其中一些更具体的领域，因为有限因子是最接近地模拟矩阵代数的算子代数。首席研究员工作的一个重要方面是博士后研究人员和博士生的培训。大多数受到首席研究员指导的年轻人现在都担任教职，他们正在培训下一代科学家和工程师。受过科学和数学训练的劳动力对于国家的技术未来至关重要，因此首席研究员将继续在年轻数学家的指导方面发挥核心作用。