Mathematical Sciences: Operator Spaces and Amenabilities

9600077阮中进研究领域是算子空间及其在算子代数、非对易调和分析和局部紧量子群中的应用。最近，利用算子空间理论，研究者研究了Kac代数和局部紧量子群的各种顺从性条件。在接下来的三年里，他计划继续这方面的研究，并计划研究以下问题：(1)Kac代数的算子顺从性、强Voular escu顺从性和Voular escu顺从性；(2)Kac代数的弱顺从性、逼近性质和弱逼近性质及其与C*-代数和von Neumann代数性质的关系；(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伯格原理：一个人必须用算符而不是函数来表示物理学的基本变量。在Heisenberg原理的启发下，数学家们研究了某些数学领域的量子化问题，如拓扑学、微分几何、分析、概率等。算子空间被定义为Hilbert空间上算子的子空间，并带有不同的矩阵范数。算子空间是函数空间的自然量子化，或者更准确地说，是Banach空间的自然量子化。算子空间理论最早是由William Arveson于1969年提出的。由于David Blecher，Edward G.Effros，Vern Paulsen，Gilles Pisier和研究人员最近的工作，它已经迅速发展成为现代分析中的一个分区。该理论在研究非自伴算子代数、C~*-代数、von Neumann代数、非对易调和分析、局部紧量子群等方面都被证明是非常有用的。在这个项目中，研究者计划进一步探索算子空间的性质和应用。S这个项目的成功将有助于我们更好地理解量子化数学。

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.