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推动的数学研究。量子力学的观点是用希尔伯特空间上的线性算子而不是标量来表示可观测量。在某些模型中，可观测量产生算子代数。时间演化由指定每个时刻代数变换的连贯规则给出。数学运算允许人们从正在建模的系统随时间演变的方式中读出其属性。在目前的工作中，将研究一种相当普遍的时间演化，称为自同态半群，目的是获得分类方案。 更具体地说，将研究与无界厄米算子的指数理论类似的开发此类半群的指数理论的问题。将寻求一种对希尔伯特空间上有界算子代数的自同态的所有伴随保持半群进行分类的方案，其中存在一个交织的等距半群。 后一种情况获得的结果将应用于超有限因子的自同态半群。