Mathematical Sciences: Endomorphisms of von Neumann Algebras

这个项目涉及数学研究， 是由量子力学激发的量子力学的观点 观点是用线性算子来表示可观测量， 希尔伯特空间而不是标量。在某些模型中， 观测量产生了一个算子代数。时间演化 是由一个连贯的规则给出的，该规则指定了 每时每刻都有代数。数学运算允许人们 从被建模的系统的方式中读出它的属性， 随着时间的推移而发展。在目前的工作中，一个相当普遍的 一种时间演化，称为自同态半群， 研究的目的是获得一个计划， 分类. 更具体地说，发展指数理论的问题 对于这样的半群，类似于指数理论， 将研究无界埃尔米特算子。的方案 的自同态的保伴半群的分类 Hilbert空间上的有界算子代数， 是一个交织半群的等距将寻求。 后一种情况下获得的结果将应用于 超有限因子的自同态半群

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.