Local Theory of Operator Spaces and Applications

算子空间局部理论及应用

• 批准号: 0140067
• 负责人: Zhong-Jin Ruan
• 金额: $ 15.76万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-05-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140067&HistoricalAwards=false
• 关键词: Local; Theory; Operator; Spaces; Applications

AbstractRuanThe operator space theory is a natural quantization of functional analysis. The major difference between operator spaces and Banach spaces is that one must consider operator matrix norms and completely bounded maps in the category of operator spaces. This was first realized by William Arveson in 1969 and was characterized by the PI in his Ph.D thesis in 1987. Since then the theory has been quickly developed into a very exciting research area in modern analysis. This remarkable development is mainly due to the contributions of D.Blecher, E.Christensen, E.Effros, M.Junge, E.Kirchberg, C.Le Merdy, G.Pisier, V.Paulsen, H.Rosenthal, R.Smith, A.Sinclair and the PI. Recently, the PI's research has been mainly centered on the 'local theory' of operator spaces and their applications. One of his main goals is to find the appropriate quantization of classical results in Banach space theory, and to apply these results to C*-algebras and von Neumann algebras, as well as to some other related areas such as non-commutative harmonic analysis and locally compact quantum groups. In this proposal, the PI plans to continue his investigation in this direction and proposes the following four research projects. (1) Investigate the local properties of non-commutative Lp spaces and their applications to operator algebras. (2) Investigate the local structure of the operator preduals of von Neumann algebras and the operator duals of C*-algebras. (3) Investigate the further applications of operator spaces to Kac algebras and locally compact quantum groups. (4) Investigate the geometric structure of the 'matrix unit balls' of operator spaces, and investigate the possible applications of operator spaces to non-commutative probability and free probability.The most profound distinction between classical and quantum mechanics is Heisenberg's principle that one should 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, which is a very important field in modern analysis. During the last fifteen years, the PI together with his colleagues has established the foundation of operator space theory and has also discovered a number of far-reaching applications to some related areas in mathematics. In this proposal, he plans to continue his work on operator spaces and their applications. He expects that the solutions the proposed research projects will make important contributions to related fields.

摘要 阮算子空间理论是泛函分析的自然量化。算子空间和 Banach 空间的主要区别在于，必须考虑算子空间范畴内的算子矩阵范数和完全有界映射。这首先由 William Arveson 于 1969 年实现，并在 1987 年的博士论文中以 PI 为特征。此后，该理论迅速发展成为现代分析中一个非常令人兴奋的研究领域。 这一显着的发展主要归功于 D.Blecher、E.Christensen、E.Effros、M.Junge、E.Kirchberg、C.Le Merdy、G.Pisier、V.Paulsen、H.Rosenthal、R.Smith、A.Sinclair 和 PI 的贡献。 最近，PI的研究主要集中在算子空间的“局部理论”及其应用上。他的主要目标之一是找到巴拿赫空间理论中经典结果的适当量化，并将这些结果应用于 C* 代数和冯诺依曼代数，以及其他一些相关领域，例如非交换调和分析和局部紧量子群。 在本提案中，PI计划继续在这个方向进行研究，并提出以下四个研究项目。 (1) 研究非交换 Lp 空间的局部性质及其在算子代数中的应用。 (2) 研究冯诺依曼代数算子预对数和C*-代数算子对偶的局部结构。 (3) 研究算子空间在Kac代数和局部紧量子群中的进一步应用。 （4）研究算子空间“矩阵单位球”的几何结构，研究算子空间在非交换概率和自由概率中的可能应用。经典力学和量子力学最深刻的区别是海森堡原理，即应该用算子而不是函数来表示物理学的基本变量。 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，是现代分析中一个非常重要的领域。在过去的十五年里，PI与他的同事一起建立了算子空间理论的基础，并在数学的一些相关领域发现了许多深远的应用。在这项提案中，他计划继续他在操作空间及其应用方面的工作。他预计所提出的研究项目的解决方案将为相关领域做出重要贡献。