Operator Algebraic Structures and Their Applications

算子代数结构及其应用

• 批准号: 9801324
• 负责人: Masamichi Takesaki
• 金额: $ 45.8万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801324&HistoricalAwards=false
• 关键词: Operator; Algebraic; Structures; Their; Applications

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).

摘要 Effros/Popa/Takesaki 三位 PI 和他们的学生将继续他们对算子代数理论的研究。 爱德华·埃夫罗斯 (Edward Effros) 正在与阮忠进 (Zhong-Jin Ruan) 合作，利用“量化”巴纳赫空间的理论，开展一项理解非交换泛函分析的重大项目。 索林·波帕 (Sorin Popa) 将继续对子因素理论的分析和组合方面进行研究。 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式，绕过麦基的不可分类性理论。 现在才变得清楚，玻尔、爱因斯坦和海森堡在本世纪初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中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，并制造具有卓越性能的新材料。 约翰·冯·诺依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。 他是第一位认识到数学是基于源自经典物理学的几何和物质的根本错误观念的数学家，因此数学家有责任开发新形式的“量子化数学”。 为此，他引入了算子代数领域，该领域现在构成了现代数学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有重要的应用。 此项资助的参与者正在追求该领域一些最令人兴奋的方向。 其中包括探索量化对称性（Popa：子因子理论）、无限维分析的量化版本（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影响（Takesaki：冯·诺依曼）代数）。