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.

摘要：算子空间是希尔伯特空间上有界线性算子的范数闭子空间，配备有区分矩阵范数。 算子空间理论是巴纳赫空间理论的自然量化。 算子空间和 Banach 空间之间的主要区别在于，我们考虑算子空间范畴中的算子矩阵范数和完全有界映射。 1987 年，PI 成功地通过矩阵范数制定了算子空间的公理化。 从那时起，这一领域取得了很大进展。 在本提案中，PI计划继续在该方向进行研究，并提出以下研究课题：（1）研究冯诺依曼代数算子预变量和$C^*$-代数算子对偶的局部结构； (2) 研究$C^*$-代数和冯诺依曼代数的局部结构； （3）研究算子空间矩阵单位球的几何结构；（4）研究在局部紧量子群中的应用。经典力学与量子力学最深刻的区别是海森堡原理，即必须用算子表示物理的基本变量。而不是函数。 J. von Neumann 的工作强调追求数学的“量化”形式非常重要。 20 世纪 40 年代，冯·诺依曼与 F.J. Murray 合作，成功地量化了积分理论。 从那时起，数学家们尝试量化许多其他数学领域，例如拓扑、微分几何、分析和概率论。 算子空间理论是泛函分析的自然量化，或者更准确地说，是巴纳赫空间理论的自然量化。 这是现代分析中最近发展的一个有前途的研究领域。 PI 和他的同事们已经为这个领域奠定了基础。 他们还发现了算子空间理论在数学相关领域的一些深远应用，例如算子代数、非交换调和分析、Kac 代数和局部紧量子群。 PI 计划继续在这个方向上进行研究，并计划探索更广泛的应用范围。