Mathematical Sciences: Endomorphisms of von Neumann Algebras

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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.