Non-commutative Analysis and Symmetry in Operator Algebra

算子代数中的非交换分析和对称性

• 批准号: 0100883
• 负责人: Masamichi Takesaki
• 金额: $ 130.27万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100883&HistoricalAwards=false
• 关键词: Non; commutative; Analysis; Symmetry; Operator

Abstract Effros/Popa/Takesaki The three PI's intend to continue their investigations on a broad range of problems in operator algebra theory. Effros and Ruan are continuing their collaboration on operator spaces. Effros and Ruan are particularly interested in studying the local theory of von Neumann algebraic preduals. Building on their earlier result with Junge that that all von Neumann preduals are locally reflexive, and that the injective preduals have a simple characterization, Effros and Ruan hope to prove that the general architecture of a von Neumann algebra can be described in terms of the local structure of the preduals. Effros also intends to look at the quantized analogues of rotundity. A number of Popa's research projects will be concerned with his axiomatization of the standard invariant for subfactors. In particular he plans to work on the most challenging problem in subfactor theory: finding techniques that would be applicable to the theory of subfactors of hyperfinite factors. He intends to continue his studies with Bisch of property T in the context of subfactors. In a very different direction, he will investigate the structure theory of C*-algebras based in part on his earlier work on the theory of local approximation by finite dimensional algebras, and of the relative Dixmier property for C*-algebras. Takesaki plans to continue his development of a canonical approach to the theory of type III factors. In the next stage of his research program, Takesaki and his collaborators intend to complete their studies of outer automorphism actions of a discrete amenable group on an approximately finite dimensional (or hyperfinite) factor of type III-lambda (lambda larger than zero) and he expects that the type III-zero case will yield to this analysis. He will continue his investigation into the most difficult problem in the area: the classfication of one-parameter groups of automorphisms. Takesaki expects to apply classfication principles learned from factor theory to the classification of certain classes of C*-algebras.Operator algebraists study the mathematics of quantum physics. In 1926 Heisenberg discovered that the paradoxes of atomic particles could be resolved with a modified version of Newtonian physics. He showed that the equations of the classical theory were still valid, provided one reinterpreted their symbols. In the classical theory these variables stand for functions. Heisenberg showed one can predict the behaviour of atomic particles if one instead regarded the variables as representing possibly infinite arrays or ``matrices'' of numbers. A few years later, von Neumann gave a mathematically precise formulation of these quantum variable in terms of Hilbert space operators. He went on to suggest that since the classical notions of measurement and geometry that underlie so much of mathematics no longer correspond to our understanding of the real world, it was necessary to seek quantized versions of mathematics. As in physics, one must begin by replacing functions by operators. In the last fifty years, operator algebraists have succeeded in quantizing a remarkable number of areas of mathematics, including analysis, topology, differential and Riemanian geometry, probability theory, and the theory of symmetry. As in quantum physics, the quantum world of mathematics is remarkable in the completely new phenomena that occur. The theory has had profound applications to various areas, including knot theory and low-dimensional topology, index theory on foliated manifolds, the classification of dynamical systems, and most recently, mathematical frameworks for both the standard model of quantum field theory (Connes) and renormalization theory (Connes and Kreimer). In this broad framework, Effros is one of the founders of quantized functional analysis (operator space theory), Popa is a leading figure in the theory of quantum symmetries (subfactor theory), and Takesaki is internationally recognized for his work on the modern theory of non-commutative integration and its use in studying the structure of von Neumann algebras and their automorphism groups

摘要Effros/Popa/Takesaki三个PI打算继续他们的调查广泛的问题在算子代数理论。Effros和Ruan正在继续他们在运营商空间方面的合作。Effros和阮特别感兴趣的是研究冯诺依曼代数预处理的局部理论。在他们早期与Junge的结果的基础上，即所有冯诺伊曼预变量都是局部自反的，并且内射预变量具有简单的特征，埃弗罗斯和阮希望证明冯诺伊曼代数的一般结构可以用局部结构来描述。预变量的结构。Effros还打算研究圆形的量子化类似物。一些波帕的研究项目将关注他的公理化的标准不变的子因素。特别是他计划工作的最具挑战性的问题，在子因子理论：寻找技术，将适用于理论的子因子的超有限因素。他打算继续他的研究与比施的财产T的背景下的子因素。在一个非常不同的方向，他将调查的结构理论的C*-代数的基础上，部分对他的早期工作理论的局部逼近有限维代数，以及相对Dixongly财产的C*-代数。竹崎计划继续他的发展规范的方法理论的第三类因素。在他的研究计划的下一个阶段，竹崎和他的合作者打算完成他们的研究外自同构行动的离散顺从组上的近似有限维（或超有限）因子的III-λ（λ大于零），他预计，III-零的情况下将产生这种分析。他将继续他的调查最困难的问题在该地区：分类的单参数群的自同构。竹崎希望将从因子理论中学到的分类原则应用于C*-代数的某些类别的分类。算子代数学家研究量子物理学的数学。1926年，海森堡发现原子粒子的悖论可以用牛顿物理学的修改版本来解决。他表明，经典理论的方程仍然有效，只要重新解释它们的符号。在经典理论中，这些变量代表函数。海森堡表明，如果人们把变量看作是可能的无限数组或“矩阵”，那么人们就可以预测原子粒子的行为。几年后，冯·诺依曼用希尔伯特空间算符给出了这些量子变量的数学精确公式。他接着提出，既然作为数学基础的经典测量和几何概念不再符合我们对真实的世界的理解，就有必要寻求数学的量子化版本。与物理学中一样，人们必须从用运算符替换函数开始开始。在过去的50年里，算子代数学家已经成功地量化了大量数学领域，包括分析、拓扑学、微分几何和黎曼几何、概率论和对称性理论。正如在量子物理学中一样，数学的量子世界在发生的全新现象中引人注目。该理论在各个领域都有着深远的应用，包括纽结理论和低维拓扑，叶状流形上的指数理论，动力系统的分类，以及最近的量子场论（Connes）和重整化理论（Connes和Kreimer）的标准模型的数学框架。在这个广泛的框架中，Effros是量子泛函分析（算子空间理论）的创始人之一，Popa是量子对称理论（子因子理论）的领军人物，Takesaki因其在现代非交换积分理论及其在研究冯诺依曼代数及其自同构群结构中的应用而获得国际认可