Mathematical Sciences: Endomorphisms of von Neumann Algebras

数学科学：冯诺依曼代数的自同态

• 批准号: 8801162
• 负责人: Robert Powers
• 金额: --
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1992-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801162&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Endomorphisms; von; Neumann

This project involves mathematical research that is motivated by quantum mechanics. The quantum mechanical point of view is to represent observable quantities by linear operators on Hilbert space rather than by scalars. In some models, the observables give rise to an algebra of operators. Time evolution is given by a coherent rule specifying a transformation of the algebra at each moment. Mathematical manipulations allow one to read off properties of the system being modeled from the way it evolves over time. In the present undertaking, a rather general sort of time evolution, called a semigroup of endomorphisms, is to be studied with the goal of obtaining a scheme of classification. More specifically, the problem of developing an index theory for such semigroups in analogy with the index theory for unbounded hermitian operators will be investigated. A scheme for classifying all adjoint-preserving semigroups of endomorphisms of the algebra of bounded operators on Hilbert space for which there is an intertwining semigroup of isometries will be sought. Results obtained for the latter situation will be applied to semigroups of endomorphisms of the hyperfinite factor.

该项目涉及由量子力学动机的数学研究。量子机械的角度是表示希尔伯特空间上的线性算子可观察到的数量，而不是标量。在某些模型中，可观察物引起了操作员的代数。时间演化是由指定代数转换的连贯规则给出的。数学操纵使人们可以从其随着时间的流逝而发展的方式读取系统的属性。在目前的工作中，应研究一种相当普遍的时间演化，称为内态性的半群，以获得分类方案的目的。 更具体地说，将研究与无界遗产运算符的索引理论开发此类半群的索引理论的问题。将寻求界定操作员代数的所有伴奏的内态分类的方案。 后一种情况获得的结果将应用于高足限因子的内态的半群。