Local Theory of Operator Spaces and Applications

摘要算子空间理论是泛函分析的自然量化。算子空间与Banach空间的主要区别在于必须考虑算子矩阵范数和算子空间范畴中的完全有界映射。这是William Arveson在1969年首次实现的，并在1987年的博士论文中以PI为特征。从那时起，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。这一显著的发展主要归功于d.b leecher、e.c ristensen、e.f effros、m.g junge、e.c kirchberg、c.l ermerdy、G.Pisier、v.p olsen、h.r onthal、R.Smith、a.c sinclair和PI的贡献。近年来，π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子化，并将这些结果应用于C*代数和冯·诺伊曼代数，以及其他一些相关领域，如非交换调和分析和局部紧致量子群。在这份建议书中，PI计划继续在这个方向进行调查，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上的应用。(2)研究von Neumann代数的算子前对偶和C*-代数的算子前对偶的局部结构。(3)进一步研究了算子空间在Kac代数和局部紧量子群中的应用。(4)研究了算子空间的“矩阵单位球”的几何结构，并研究了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之间最深刻的区别是海森堡的原理，即人们应该用算子而不是函数来表示物理的基本变量。冯·诺伊曼的工作强调了追求数学“量子化”形式的重要性。在20世纪40年代，冯·诺伊曼与f·j·默里合作，成功地将积分理论量子化。从那时起，数学家们试图量化许多其他数学领域，如拓扑、微分几何、分析和概率论。算符空间理论是泛函分析的自然量化，是现代分析的一个重要领域。在过去的十五年中，PI与他的同事们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。在这个提案中，他计划继续他在算子空间及其应用方面的工作。他期望所提出的研究项目的解决方案将对相关领域作出重要贡献。

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.