Local Theory of Operator Spaces and Applications

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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.