Local Theory of Operator Spaces and Applications

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中的算子矩阵范数和完全有界映射。这是由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，该理论已迅速发展成为现代分析中一个非常令人兴奋的研究领域。 这一引人注目的发展主要归功于D.Blecher、E.Christensen、E.Effros、M.Junge、E.基希贝格、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。他的主要目标之一是找到适当的量化的经典结果在Banach空间理论，并将这些结果应用到C*-代数和冯诺依曼代数，以及其他一些相关领域，如非交换调和分析和局部紧量子群。 在这份建议书中，PI计划继续这方面的研究，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)研究了vonNeumann代数的算子预算子和C ~*-代数的算子预算子的局部结构。 (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)研究算子空间的“矩阵单位球”的几何结构，并研究算子空间对非交换概率和自由概率的可能应用。经典力学和量子力学之间最深刻的区别是海森堡的原则，即应该用算子而不是函数来表示物理学的基本变量。冯·诺依曼的工作强调了追求数学的“量子化”形式的重要性。冯·诺依曼与F. J.默里合作，在20世纪40年代成功地量化了积分理论。从那时起，数学家们试图探索数学的许多其他领域，如拓扑学，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，泛函分析是现代分析中一个非常重要的领域。在过去的十五年里，PI和他的同事们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中发现了一些意义深远的应用。在这份提案中，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能为相关领域做出重要贡献.

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.