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.
