Vershik, A. M. 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 Cited in 1 ReviewCited in 3 Documents 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 Keywords:measurable realization of the group of positive unitary operators; universal linearization; free measure; integral representations of positive operators; positive Markov contraction; polymorphism PDFBibTeX XMLCite \textit{A. M. Vershik}, Sib. Math. J. 28, No. 1--2, 36--43 (1987; Zbl 0627.47012); translation from Sib. Mat. Zh. 28, No. 1(161), 52--60 (1987) Full Text: DOI References: [1] L. V. Kantorovich, B. Z. Vulikh, and A. G. Pinsker, Functional Analysis in Partially Ordered Spaces [in Russian], GITTL, Moscow (1950). · Zbl 0037.07201 [2] A. M. Vershik, ?A measurable realization of continuous groups of automorphisms of a unitary ring,? Izv. Akad. Nauk SSSR, Ser. Mat.,29, No. 1, 127-136 (1965). [3] G. W. Mackey, ?Point realizations of transformation groups,? J. Math.,6, No. 2, 327-335 (1962). · Zbl 0178.17203 [4] G. Maruyama, ?Transformations of flows,? J. Math. Soc. Jpn.,18, No. 3, 303-330 (1966). · Zbl 0166.40402 [5] A. Ramsay, ?Topologies on measured groupoids,? J. Funct. Anal.,47, No. 3, 314-342 (1982). · Zbl 0519.22003 [6] A. M. Vershik, ?Approximation in measure theory,? Abstract of Doctoral Dissertation, Phys.-Math. Sciences, Leningrad State Univ. (1973). [7] V. A. Rokhlin, ?Unitary rings,? Dokl. Akad. Nauk SSSR,59, No. 4, 643-646 (1948). [8] G. I. Kats, ?Generalized functions on a locally compact group, and decompositions of unitary representations,? Trudy Mosk. Mat. Obshch.,10, 3-40 (1961). [9] A. M. Vershik and A. L. Fedorov, ?Trajectory theory,? Itogi Nauki i Tekhniki, Tom 26, VINITI, Moscow (1985). [10] A. M. Vershik, ?Multivalued maps with an invariant measure (polymorphisms) and Markov processes,? J. Sov. Math.,23, No. 3 (1983). [11] A. L. Fedorov, ?Polymorphisms and partitions of Lebesgue spaces,? Funkts. Anal. Prilozhen.,16, No. 2, 88-89 (1982). [12] J. Renault, ?A groupoid approach to C*-algebras,? Lect. Notes Math., No. 793, Springer-Verlag, New York (1980). · Zbl 0433.46049 [13] G. E. Shilov and Fan Dyk Tin’, Integral, Measure and Derivative in Linear Spaces [in Russian], Nauka, Moscow (1967). [14] S. V. Fomin, ?On measures invariant under a certain group of transformations,? Izv. Akad. Nauk SSSR, Ser. Mat.,14, No. 3, 261-274 (1950). · Zbl 0037.07501 [15] S. Albeverio, R. Hoegh-Krohn, D. Testard, and A. M. Vershik, ?Factorial representations of path groups,? J. Funct. Anal.,51, No. 1, 115-131 (1983). · Zbl 0522.22013 [16] A. M. Vershik, ?The Markov processes from the standpoint of the theory of contraction in linear and complex analysis,? Lecture Notes Math., No. 1043, 101-103, Springer-Verlag, New York (1984). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.