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/takeaki这三个PI打算继续研究操作员代数理论中广泛的问题。 Effros和Ruan继续在操作员空间上进行合作。 EFFROS和RUAN对研究von Neumann代数预期的局部理论特别感兴趣。基于朱吉（Junge）的早期结果，即所有冯·诺伊曼（Von Neumann）的预期都是局部反思性的，并且注入式预期具有简单的特征，Effros和Ruan希望证明可以用von Neumann代数的一般体系结构来描述，以预期的局部结构来描述。 EFFROS还打算查看量子的量化类似物。许多Popa的研究项目将与他对子因子的标准不变型的公理化有关。特别是他计划在亚比例理论中解决最具挑战性的问题：寻找适用于高铁因素的亚比例理论的技术。他打算在子因子的背景下继续使用财产t进行研究。在一个非常不同的方向上，他将研究基于有限维代数的局部近似理论的C* - 代数理论，以及C*-Algebras的相对dixmier属性。 Takeaki计划继续开发针对III型因素理论的规范方法。在他的研究计划的下一个阶段，Takeaki和他的合作者打算在大约有限的尺寸（或高限度）IIIII-LAMBDA（LAMBDA大于零）的尺寸（或高限度）上完成对离散木材的外部自动形态作用的研究，他预计III-Zero型案例将产生该分析。他将继续对该领域最困难的问题进行调查：单态群体的分类。 Takeaki希望将从因子理论学到的分类原则应用于某些类别的C*-Algebras的分类。操作者代数研究量子物理学的数学。海森伯格（Heisenberg）在1926年发现，可以通过修改后的牛顿物理学来解决原子颗粒的悖论。他表明，古典理论的方程式仍然有效，只要有人重新解释了它们的符号。在经典理论中，这些变量代表函数。海森伯格（Heisenberg）表明，如果一个人认为变量代表了可能的无限阵列或数字的“矩阵”，则可以预测原子粒子的行为。几年后，冯·诺伊曼（von Neumann）用希尔伯特（Hilbert）空间运营商对这些量子变量进行了数学精确的配方。他继续暗示，由于基于许多数学的经典测量和几何观念不再与我们对现实世界的理解相对应，因此有必要寻求量化的数学版本。与物理学一样，必须首先替换操作员的功能。在过去的五十年中，运营商代数主义者成功地量化了许多数学领域，包括分析，拓扑，差异和侵蚀性几何学，概率理论和对称性理论。与量子物理学一样，数学的量子世界在发生的全新现象中是显着的。该理论在各个领域都有深刻的应用，包括结理论和低维拓扑，叶状歧管的索引理论，动态系统的分类以及最近的数学框架，用于量子场理论（CONNES）的标准模型（Connes and Kreimer）。在这个广泛的框架中，EFFROS是量化功能分析（操作员空间理论）的创始人之一，Popa是量子对称理论（亚比例理论）的领先人物，而Takeaki在国际上因其在现代的非交通整合理论及其在研究Voneumann Altgebras及其群体结构中的使用而受到国际认可。