STUDIES ON FUNCTIONAL AND REAL ANALYSES VIA UNIVERSAL AND UNIFIED APPROACH

研究结果的详细内容（包括贡献者姓名）请参见“研究报告”;我们将仅说明每个研究小组通过项目主要关注的研究对象。（1）（算子代数和函数代数研究组）研究了II_1因子、完全正映射、非交换动力系统、Kac群和量子群的指标理论（算子代数理论），以及道格拉斯代数、Bourgain代数、Toeplitz和Hankel算子等（函数代数理论），给出了许多重要结果。（2）（Banach函数空间研究组）非对易过程（其系统的稳定性）、克拉克森型不等式、（线性、半线性和非线性）演化方程、量子系统的熵（在遗传理论中的应用）、非线性Perron-Frobenius理论、外函数等，从不同的角度进行了研究，并取得了许多成果。（3）（研究 关于我们 表示论上的群）量子群的表示、无限维李群和李代数及其表示、Kostant理论和Feynman路径积分、半单李群的齐性空间等，被广泛调查。（4）（偏微分方程研究组）算子半群与发展方程、界面的渐近性态与非线性方程解的爆破、薛定谔算子的散射理论、量子场论的数学研究、WKB方法、微局部分析、Torotter型公式等都被研究过了，大大增加了我们的理论知识。（5）（真实的分析研究组）研究Federer测度的整体密度定理、局部凸空间中的Sato广义函数、Festiman-Phong不等式、交换变换群的拟不变测度、无限测度空间的外推等。的目的和结果，似乎是有影响力的。少

Details of results (including names of contributors) should be found in "THE RESEARCH REPORT"; We will state only the research objects with which each research group has mainly concerned through the project.(1)(Research group on operator algebras and function algebras) Index theory for II_1 factors, completely positive maps, non-abelian dynamical systems, Kac groups and quantum groups (Operator algebra theory), and Douglas algebras, Bourgain algebras, Toeplitz and Hankel operators, etc. (Function algebra theory), are investigated in this group; many results which seem to be important arer given.(2)(Research group on Banach function spaces) Non-commutative processes (stability of their system), Clarkson type inequality, (linear, semi-linear, and non-linear) evolution equation, entropies of quantum systems (with applications to heredity theory), non-linear Perron-Frobenius theory, outer functions, etc., are investigated from various stand points and many results are obtained.(3)(Research … More group on representation theory) Representations of quantum groups, infinite-dimensional Lie groups, and Lie algebras and their representations, Kostant theory and Feynman path integral, homogeneous spaces of semi-simple Lie groups, etc., were investigated extensively.(4)(Research group on partial differential equations) Semigroups of operators and evolution equation, asymptotic behavior of interfaces and blow-up of solutions for non-linear equations, Scattering theory for Schrodinger operators, mathematical study of quantum field theory, WKB method,microlocal analysis, Torotter type formulas, etc., were studied and added greately to our knowledge of the theory.(5)(Research group on real analysis) Studies on global density theorem for Federer measures, Sato's generalized functions valued in locally convex spaces, Fefferman-Phong inequality, quasi-invariant measures for commutative transformation groups, extrapotation for infinite measure spaces, etc., are aimed and results which seem to be influential are obtained. Less