Mathematical Sciences: Operator Spaces and Amenabilities

9600077阮仲金主要研究方向为算子空间及其在算子代数、非交换调和分析和局部紧致量子群中的应用。近年来，研究者利用算子空间理论研究了Kac代数和局部紧量子群的各种适应条件。在接下来的三年里，他计划继续这个方向的研究，并计划研究以下问题：(1)Kac代数的算子适性、强Voiculescu适性和Voiculescu适性；(2) Kac代数的弱适应性、逼近性和弱逼近性，以及它们与C*代数和von Neumann代数性质的联系；(3)局部紧量子群。经典力学和量子力学之间最深刻的区别是海森堡原理：人们必须用算子而不是函数来表示物理的基本变量。在海森堡原理的激励下，数学家们研究了某些数学领域的量化，如拓扑、微分几何、分析、概率论等。一个算子空间被定义为Hilbert空间上具有不同矩阵范数的算子的子空间。算子空间是函数空间的自然量化，或者更准确地说，是巴拿赫空间的自然量化。算子空间理论是由William Arveson在1969年首次提出的。由于David Blecher， Edward G. Effros, Vern Paulsen， Gilles Pisier和研究者最近的工作，它已经迅速发展成为现代分析的一个分支领域。该理论在非自伴随算子代数、C*-代数、von Neumann代数、非交换调和分析、局部紧量子群等方面的研究中具有重要的应用价值。在这个项目中，研究者计划进一步研究算子空间的性质和应用。这个项目的成功将对我们更好地理解量化数学有很大的帮助。

9600077 Zhong-Jin Ruan The investigator's research area is in operator spaces and their applications to operator algebras, non-commutative harmonic analysis and locally compact quantum groups. Recently, using operator space theory, the investigator has studied various amenability conditions for Kac algebras and locally compact quantum groups. During next three years, he plans to continue his research in this direction and plans to investigate the following problems: (1) Operator amenability, strong Voiculescu amenability and Voiculescu amenability for Kac algebras; (2) Weak amenability, approximation property and weak approximation property for Kac algebras, and their connection with C*-algebra and von Neumann algebra properties; (3) Locally compact quantum groups. The most profound distinction between classical and quantum mechanics is Heisenberg's principle: one must represent the basic variables of physics by operators rather than functions. Motivated by Heisenberg's principle, mathematicians have investigated the quantization of certain areas of mathematics such as topology, differential geometry, analysis, probability and etc. An operator space is defined to be a subspace of operators on a Hilbert space together with a distinct matrix norm. Operator spaces are natural quantization of function spaces, or more precisely, natural quantization of Banach spaces. The theory of operator spaces was first introduced by William Arveson in 1969. It has been quickly developed into a subarea in modern analysis due to the recent work of David Blecher, Edward G. Effros, Vern Paulsen, Gilles Pisier and the investigator. The theory has been proved to be extremely useful in the study of non-self adjoint operator algebras, C*-algebras, von Neumann algebras, non-commutative harmonic analysis, locally compact quantum group, and etc. In this project, the investigator plans to pursue the further properties and applications of operator spaces. The succes s of this project will be very helpful for us to have a better understanding of the quantized mathematics.