Operator Spaces and Their Applications

算子空间及其应用

• 批准号: 9877157
• 负责人: Zhong-Jin Ruan
• 金额: $ 11.1万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-05-15 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9877157&HistoricalAwards=false
• 关键词: Operator; Spaces; Their; Applications

AbstractRuanAn operator space is a norm closed subspace of bounded linear operators on a Hilbert space, equipped with a distinguished matrix norm. The operator space theory is a natural quantization of Banach space theory. The major difference between operator spaces and Banach spaces is that one considers operator matrix norms and completely bounded maps in the category of operator spaces. In 1987, the PI succeeded in formulating an axiomatization of operator spaces by matrix norms. Since then, a lot of progress has been made in this area. In this proposal, the PI plans to continue his research in this direction and proposes the following research topics:(1) investigate the local structure of the operator preduals of von Neumann algebras and the operator duals of $C^*$-algebras; (2) investigate the local structure on $C^*$-algebras and von Neumann algebras; (3) investigate the geometric structure of matrix unit balls of operator spaces;(4) investigate the application to locally compact quantum groups.The most profound distinction between classical and quantum mechanics is Heisenberg's principle that one must represent the basic variables of physics by operators rather than functions. The work of J. von Neumann emphasized that it is important to pursue the "quantized" forms of mathematics. Collaborating with F.J. Murray, von Neumann succeeded in quantizing integration theory during the 1940's. Since then, mathematicians have tried to quantize many other areas of mathematics such as topology, differential geometry, analysis and probability theory. The theory of operator spaces is a natural quantization of functional analysis, or more precisely, a natural quantization of Banach space theory. This is a recently developed promising research area in modern analysis. The PI and his colleagues have established the foundation of this area. They have also discovered some far-reaching applications of operator space theory to related areas in mathematics such as operator algebras, non-commutative harmonic analysis, Kac algebras and locally compact quantum groups. The PI plans to continue his research in this direction and plans to explore a much broarder range of applications.

Abstractruanan操作员空间是一个有界线性算子在希尔伯特空间上的范围封闭子空间，配备了杰出的矩阵标准。 操作员空间理论是Banach空间理论的自然量化。 操作员空间和BANACH空间之间的主要区别在于，一个人认为操作员矩阵规范和操作员空间类别中的完全有限的地图。 1987年，PI成功地通过矩阵规范制定了操作员空间的公理化。 从那时起，该领域已经取得了很多进展。 在此提案中，PI计划继续朝这个方向进行研究，并提出以下研究主题：（1）调查von Neumann代数的运营商预期的本地结构和$ C^*$ - 代数的操作员双重； （2）研究$ C^*$ - 代数和冯·诺伊曼代数的本地结构； （3）研究操作员空间的基质单位球的几何结构；（4）研究对局部紧凑型量子组的应用。经典和量子力学之间最深刻的区别是海森伯格的原理，即一个人必须代表操作员而不是功能而不是功能的物理学基本变量。 J. von Neumann的工作强调，追求“量化”数学形式很重要。 冯·诺伊曼（Von Neumann）与F.J. Murray合作，在1940年代成功地量化了整合理论。 从那以后，数学家试图量化许多其他数学领域，例如拓扑，差异几何，分析和概率理论。 操作员空间的理论是对功能分析的自然量化，或更确切地说是Banach空间理论的自然量化。 这是一个最近在现代分析中开发的有希望的研究领域。 PI和他的同事已经建立了该地区的基础。 他们还发现了操作员空间理论对数学相关领域的一些深远应用，例如操作员代数，非交通谐波分析，KAC代数和本地紧凑的量子组。 PI计划继续朝这个方向进行研究，并计划探索许多范围的应用程序。