Operator Algebraic Structures and Their Applications

摘要Effros/Popa/Takesaki 三个PI的和他们的学生将继续他们的调查算子代数理论。 爱德华·埃弗罗斯正在着手与阮仲金合作，利用“量化”Banach空间理论来理解非交换泛函分析的主要计划。 索林波帕将继续他的调查子因子理论在其分析和组合方面。 Masamichi Takesaki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。 这项工作使他能够制定一个范式分类问题的数学绕过不可分类理论的麦基。 直到现在，人们才逐渐清楚，玻尔、爱因斯坦和海森堡在本世纪早期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。 特别是，我们将能够利用量子力学经典的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，并制造具有显着特性的新材料。 这些可能性在某种程度上被约翰·冯·诺依曼预见到了，他为量子力学发明了正确的数学语言。 他是第一个数学家认识到，因为数学是基于从根本上错误的想法几何和问题所产生的经典物理学，这是义不容辞的数学家发展新形式的“量化数学”。 为此目的，他介绍了该地区的运营商代数，现在构成了一个最令人兴奋的分支现代数学。 这门学科在几何学、代数学、概率论和数学物理学中有重要的应用。 该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。 这些包括探索量化对称性（波帕：子因子理论），量化版本的无限维分析（Effros：运营商空间），并了解分类问题的运营商代数及其深刻的影响，这些问题在整个数学（竹崎：冯诺依曼代数）。

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).