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Ukrains’kyi Matematychnyi Zhurnal

Об одном условии регулярности квантовых квадратичных стохастических процессов

Farrukh Maksutovich Mukhamedov, Ф. М. Мухамедов, Ф. М. Мухамедов

Published Oct 25, 2001• 0 citations
OpenAlex

Abstract

We present necessary and sufficient conditions for the validity of a regularity condition for homogeneous quantum quadratic stochastic processes defined on von Neumann algebras.

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Von Neumann algebras associated with the normal representation of canonical commutation relations are studied. Corresponding to each subspace of a real Hilbert space (test function space), a von Neumann algebra on another complex Hilbert space (the Fock space) is defined. This correspondence is proved to be an isomorphism between a certain complemented lattice of subspaces and that of the von Neumann algebras. This result has an application to the duality theorem in the theory of a free scalar field, which is to be discussed in a separate paper. A necessary and sufficient condition on a subspace, in order that the corresponding von Neumann algebra is of type I, is obtained.