Noncommutative function theory in operator algebras and operator spaces

算子代数和算子空间中的非交换函数论

• 批准号: 1201506
• 负责人: David Blecher
• 金额: $ 8.63万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201506&HistoricalAwards=false
• 关键词: Noncommutative; function; theory; operator; algebras

The principal investigator proposes two main lines of research, focused around new directions, and around foundational problems, in the relatively new field of operator spaces. These lines are: 1) 'noncommutative functional analysis and noncommutative Banach function theory', which in part continues the investigators introduction, and application, of powerful tools from classical functional analysis and C*-algebra and von Neumann algebra theory to 'noncommutative functional analysis', 2) the general theory and structure of operator algebras, for example extending to general operator algebras important new notions of equivalence that are attracting widespread attention in the field. The investigator and collaborators are also proposing to investigate applications of the above.The study of operator algebras originally grew out of quantum mechanics. It is often of crucial importance to see how formulas involving numerical variables behave when these variables are allowed to be operator variables. Because operator variables do not commute, this is often called 'noncommutative mathematics' It is out of such a process that the theory of operator spaces and completely bounded maps emerged. This theory, which the investigator helped to found, is a novel and compelling approach to problems involving linear analysis which arise in 'noncommutative mathematics'. The investigator's research focuses partly on transferring important ideas and tools from classical subfields of linear analysis, via such 'quantization', to solve significant problems in 'noncommutative' mathematics. Most of the projects involve a deep unification of ideas from several different research areas; and often require extremely deep and technically demanding analysis. Successful findings would bring several subjects closer together, attract scientists with disparate backgrounds, and lead to cross-fertilization between these disciplines.

首席研究人员提出了两条主要的研究线，集中在新方向和基础问题周围，在相对较新的操作员空间领域。 这些行是：1）“非共同的功能分析和非共同的BANACH函数理论”，部分继续进行研究者的介绍和应用，从经典的功能分析和C*-Algebra和Von Neumann代数理论到“非交换性功能分析”的强大工具，例如，对经营者的一般理论和新范围的一般性结构，2）在该领域引起广泛的关注。研究人员和合作者还提议研究上述应用的应用。对代数的研究最初是从量子力学中生长的。查看涉及数值变量的公式如何在允许这些变量为操作器变量时表现出至关重要的重要性。由于运算符变量不通勤，因此通常称为“非交通性数学”，因此出于一个过程，以至于运算符空间理论和完全有限的地图出现。 研究者帮助发现的这一理论是一种新颖而令人信服的方法，涉及“非共同数学”中出现的线性分析的问题。研究者的研究部分重点是通过这种“量化”从线性分析的古典子字段转移重要的思想和工具，以解决“非共同”数学中的重大问题。大多数项目都涉及从几个不同研究领域的思想深入统一。通常需要非常深入且技术要求的分析。 成功的发现将使几个主题更加紧密，吸引具有不同背景的科学家，并导致这些学科之间的交叉施用。