Operator Algebraic Structures and Their Applications

Effros/Popa/Takesaki这三位PI和他们的学生将继续研究算子代数理论。Edward Effros正在与阮仲金（Zhong-Jin Ruan）合作，利用“量子化”巴拿赫空间的理论，着手一个理解非交换泛函数分析的重大项目。Sorin Popa将继续他在子因子理论的分析和组合方面的研究。Masamichi Takesaki将利用他和他的合作者在寻找冯·诺伊曼代数的内在基础结构方面取得的巨大进展来取得进一步的进展。这项工作使他能够为数学中的分类问题制定一个范例，绕过了麦基的不可分类性理论。波尔、爱因斯坦和海森堡在本世纪初所思考的量子物理学的深奥概念将对现代技术产生深远的影响，这一点直到现在才变得清晰起来。特别是，我们将能够利用量子力学的经典不可思议的预测来建造新型计算机，以前所未有的精度测量物理量，并制造具有非凡性能的新材料。约翰·冯·诺伊曼在某种程度上预见到了这些可能性，他发明了量子力学的正确数学语言。他是第一个认识到，由于数学是建立在经典物理学中产生的几何学和物质的基本错误观念的基础上的，数学家有责任发展新的“量子化数学”形式。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。这项资助的参与者正在追求这个领域中一些最令人兴奋的方向。其中包括对量子化对称的探索（Popa：子因子理论），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对整个数学中此类问题的深刻含义（Takesaki：冯·诺伊曼代数）。

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).