Operator Spaces and Their Applications

算子空间是Hilbert空间上有界线性算子的范数闭子空间，具有一个可分辨的矩阵范数。算子空间理论是巴拿赫空间理论的自然量子化。算子空间与巴拿赫空间的主要区别在于在算子空间的范畴中考虑算子矩阵范数和完全有界映射。1987年，PI通过矩阵范数成功地表述了算子空间的公理化。从那时起，这一领域取得了很大进展。在本论文中，PI计划在此方向继续研究，并提出以下研究课题：(1)研究von Neumann代数的算子前对偶和$C^*$-代数的算子对偶的局部结构；(2)研究了$C^*$-代数和von Neumann代数的局部结构；(3)研究了算子空间的矩阵单位球的几何结构；(4)研究了该方法在局部紧量子群中的应用。经典力学和量子力学之间最深刻的区别是海森堡的原理，即必须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的“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子化，或者更准确地说，是巴拿赫空间理论的自然量子化。这是现代分析中最近发展起来的一个有前途的研究领域。PI和他的同事已经建立了这个领域的基础。他们还发现了算子空间理论在数学相关领域的一些深远的应用，如算子代数、非交换调和分析、Kac代数和局部紧致量子群。PI计划在这个方向上继续他的研究，并计划探索更广泛的应用范围。

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.