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Compactness of the congruence group of measurable functions in several variables

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Abstract

We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let \(f:\prod\limits_{i = 1}^n {X_i \to Z} \) be a measurable function defined on the product of finitely many standard probability spaces (Xi, \(\mathfrak{B}_i \), μi), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g1, ..., gn) of measure preserving automorphisms of the respective spaces (Xi, \(\mathfrak{B}_i \), μi) such that f(g1 x 1, ..., gnxn) = f(x1, ..., xn) almost everywhere and prove that this group is compact, provided that its “trivial” symmetries are factored out. As a consequence, we are able to characterize all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which was solved by the first author but is interesting in itself. Bibliography: 5 titles.

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References

  1. V. S. Varadarajan, Geometry of Quantum Theory. Vol. II, The University Series in Higher Mathematics, Van Nostrand Reinhold Co., New York-Toronto, Ontario-London (1970).

    MATH  Google Scholar 

  2. A. M. Vershik, “Classification of measurable functions of several arguments, and invariantly distributed random matrices,” Funkts. Anal. Prilozhen., 36, No. 2, 12–27 (2002)

    MathSciNet  Google Scholar 

  3. A. M. Vershik, “The universal Uryson space, Gromov’s metric triples, and random metrics on the series of natural numbers,” Russian Math. Surveys, 53, No. 5, 57–64 (1998).

    MathSciNet  Google Scholar 

  4. A. M. Vershik, “Measure theoretic constructions and their applications in ergodic theory, asymptotics, combinatorics, and geometry,” Lecture notes for a course held in Autumn 2002 at the Erwin Schrödinger Institute (Vienna). To appear in the ESI Lecture Notes Series, published by the European Mathematical Society.

  5. R. J. Zimmer, Ergodic Theory and Semisimple Groups (Monographs in Mathematics, 81), Birkhäuser Verlag, Basel (1984).

    MATH  Google Scholar 

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Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St.Peterburg, Russia

    A. M. Vershik

  2. Faculty of Mathematics, University of Vienna, Austria

    U. Haböck

Authors
  1. A. M. Vershik
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  2. U. Haböck
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 57–67.

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Vershik, A.M., Haböck, U. Compactness of the congruence group of measurable functions in several variables. J Math Sci 141, 1601–1607 (2007). https://doi.org/10.1007/s10958-007-0068-7

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  • DOI: https://doi.org/10.1007/s10958-007-0068-7

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