Operator Algebraic Structures and Their Applications

摘要Effros/Popa/Takesaki三位PI和他们的学生将继续他们在算子代数理论方面的研究。爱德华·埃夫罗斯正在与阮仲晋合作，利用“量子化”的Banach空间理论，开展一个理解非对易泛函分析的重大项目。索林·波帕将继续他在分析和组合方面对子因子理论的研究。武崎正道将利用他和他的合作者在寻找冯·诺依曼代数的内在基础结构方面取得的巨大进展，取得进一步的进展。这项工作使他能够为数学中的分类问题制定一种范式，绕过麦基的不可分类理论。直到现在才清楚的是，玻尔、爱因斯坦和海森伯格在本世纪初所思考的深奥的量子物理概念，将对现代技术产生深远的影响。特别是，我们将能够利用量子力学在分类上不可想象的预测来建造新型计算机，以前所未有的精度测量物理量，并制造具有显著性能的新材料。约翰·冯·诺伊曼在某种程度上预见到了这些可能性，他为量子力学发明了正确的数学语言。他是第一个认识到这一点的数学家，因为数学是建立在对几何和物质的根本错误认识基础上的，这是数学家义不容辞的责任，开发新的形式的“量子化数学”。为此，他引入了算子代数领域，这一领域现在构成了现代数学中最令人兴奋的分支之一。这门学科在几何、代数、概率论和数学物理中都有重要的应用。这笔赠款的参与者正在追寻这一领域最令人兴奋的方向。其中包括对量子化对称性的探索(POPA：子因子论)，无限维分析的量子化版本(Effros：算子空间)，以及理解算子代数的分类问题及其在整个数学中对此类问题的深刻含义(Takesaki：von Neumann代数)。

Abstract Effros/Popa/Takesaki The three PI's and their students will continue their investigations in operator algebra theory. Edward Effros is embarking on a major program of understanding non-commutative functional analysis in collaboration with Zhong-Jin Ruan, using the theory of "quantized" Banach spaces. Sorin Popa will be continuing his investigation of subfactor theory in both its analytic and combinatorial aspects. Masamichi Takesaki will be taking advantage of the enormous progress that he and his collaborators have made in finding the intrinsic underlying structure of von Neumann algebras to make further progress. This work has enabled him to formulate a paradigm for classification problems in mathematics that bypasses the non-classifiability theory of Mackey. It is only now becoming clear that the abstruse notions of quantum physics pondered by Bohr, Einstein, and Heisenberg in the early part of this century, will have profound effect on modern technology. In particular, we will be able to exploit the classifically inconceivable predictions of quantum mechanics to build new kinds of computers, measure physical quantities with unprecedented accuracy, and make new materials with remarkable properties. These possibilities were to some degree foreseen by John von Neumann, who invented the correct mathematical language for quantum mechanics. He was the first mathematician to realize that since mathematics is based on fundamentally mistaken ideas of geometry and matter arising from classical physics, it was incumbent on mathematicians to develop new forms of "quantized mathematics". For that purpose he introduced the area of operator algebras, which now constitutes one of the most exciting branches of modern mathematics. This subject has important applications to geometry, algebra, probability theory, and mathematical physics. The participants in this grant are pursuing some of the most exciting directions in this field. These include the exploration of quantized symmetry (Popa: subfactor theory), the quantized version of infinite dimensional analysis (Effros: operator spaces), and understanding the classification problems of operator algebra and its deep implications for such problems throughout mathematics (Takesaki: von Neumann algebras).