Non-commutative Analysis and Symmetry in Operator Algebra

摘要Effros/Popa/Takesaki三位Pi打算继续他们对算子代数理论中广泛问题的研究。Effros和Ruan正在继续他们在操作员空间的合作。Effros和Ruan对研究von Neumann代数乘积的局部理论特别感兴趣。Effros和Ruan在Junge得到的所有von Neumann前件都是局部自反的结果的基础上，证明了von Neumann代数的一般结构可以用前件的局部结构来刻画。埃夫罗斯还打算研究圆形的量化类比。波帕的一些研究项目将涉及他对子因子的标准不变量的公理化。特别是，他计划致力于子因子论中最具挑战性的问题：寻找适用于超有限因子子因数理论的技术。他打算在子因子的背景下继续他与性质T的Bisch的研究。在一个非常不同的方向上，他将研究C*-代数的结构理论，部分基于他早期关于有限维代数的局部逼近理论的工作，以及C*-代数的相对Dixmier性。武崎计划继续发展他对第三类因素理论的规范方法。在他的研究计划的下一阶段，Takesaki和他的合作者打算完成他们对离散顺从群在类型III-lambda(λ大于零)的近似有限维(或超有限)因子上的外自同构作用的研究，他预计类型III-0的情况将导致这种分析。他将继续研究该领域最困难的问题：单参数自同构群的分类。Takesaki希望将从因子论中学到的分类原则应用于某些类别的C*-代数的分类。算子代数学家研究量子物理的数学。1926年，海森伯格发现原子粒子的悖论可以用牛顿物理学的修正版本来解决。他证明了经典理论的方程式仍然有效，只要人们重新解释它们的符号。在经典理论中，这些变量代表函数。海森伯格指出，如果人们将变量视为可能代表无限数组或数字的“矩阵”，就可以预测原子粒子的行为。几年后，冯·诺伊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。他接着表示，由于构成如此多数学基础的经典测量和几何概念不再符合我们对现实世界的理解，因此有必要寻求数学的量化版本。就像在物理学中一样，必须从用运算符替换函数开始。在过去的五十年里，算符代数学家成功地量化了许多数学领域，包括分析、拓扑学、微分和黎曼几何、概率论和对称论。就像在量子物理学中一样，量子数学世界在出现的全新现象中是引人注目的。这一理论在许多领域都有广泛的应用，包括纽结理论和低维拓扑、层状流形上的指数理论、动力系统的分类，以及最近建立的量子场论标准模型(Connes)和重整化理论(Connes和Kreimer)的数学框架。在这个广泛的框架中，Effros是量子化泛函分析(算符空间理论)的创始人之一，Popa是量子对称理论(子因子论)的领军人物，Takesaki因其在现代非对易积分理论方面的工作及其在研究von Neumann代数及其自同构群的结构中的应用而得到国际公认

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