Mathematical Sciences: Endomorphisms of von Neumann Algebras

这个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，可观测量产生了算子代数。时间演化是由一个连贯的规则给出的，该规则规定了在每个时刻的代数变换。数学操作允许人们从系统随时间演变的方式中读出被建模系统的属性。在目前的工作中，研究一种相当一般的时间演化，称为自同态的半群，目的是获得一种分类方案。更具体地说，我们将研究类似于无界厄米算子的指标理论的关于这类半群的指标理论。寻求Hilbert空间上有界算子代数的所有伴保自同态半群的一种分类方案，该半群存在一个交织的等距半群。后一种情况下得到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.