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Measurable realizations of groups of automorphisms, and integral representations of positive operators. (English. Russian original) Zbl 0627.47012

Sib. Math. J. 28, No. 1-2, 36-43 (1987); translation from Sib. Mat. Zh. 28, No. 1(161), 52-60 (1987).
The paper is dedicated to the memory of L. V. Kantorovich and is connected with some of his mathematical ideas. In the first section the author is concerned with the measurable realization of the group of positive unitary operators. Although this problem of ergodic theory has been solved, the author improves an older solution of this, giving explicitly the universal linearization by means what is called a free measure. The next section contains results related to integral representations of positive operators. He obtains, in particular, an integral representation for every positive Markov contraction, for which the concept of polymorphism is needed. The third and last section contains examples, counterexamples and some open problems.
Reviewer: F.-H.Vasilescu

MSC:

47B38 Linear operators on function spaces (general)
47B60 Linear operators on ordered spaces
47D03 Groups and semigroups of linear operators
28D99 Measure-theoretic ergodic theory
47A35 Ergodic theory of linear operators
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