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Duality and Free Measures in Vector Spaces, the Spectral Theory of Actions of Non-Locally Compact Groups

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The paper presents a general duality theory for vector measure spaces taking its origin in author’s papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications.

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References

  1. S. Albeverio, B. K. Driver, M. Gordina, and A. Vershik, “Equivalence of the Brownian and energy representations,” Zap. Nauchn. Semin. POMI, 441, 17–44 (2015).

    MathSciNet  MATH  Google Scholar 

  2. S. Albeverio, R. Hoegh-Krohn, D. Testard, and A. Vershik, “Factorial representations of path groups,” J. Funct. Anal., 51, 115–131 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  3. N. K. Bary, Trigonometric Series, The Macmillan Co., New York (1964).

    MATH  Google Scholar 

  4. A. Connes, J. Feldman, and B. Weiss, “An amenable equivalence relation is generated by single transformation,” Ergodic Theory Dynam. Systems, 1, 431–450 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  5. L. N. Dovbysh and V. N. Sudakov, “On the Gram–de Finetti matrices,” Zap. Nauchn. Semin. LOMI, 119, 77–86 (1982).

    MathSciNet  MATH  Google Scholar 

  6. I. M. Gelfand, M. I. Graev, and A. M. Vershik, “Representations of the group of functions taking values in a compact Lie group,” Compos. Math., 42, 217–243 (1980).

    MathSciNet  MATH  Google Scholar 

  7. I. M. Gelfand, M. I. Graev, and A. M. Vershik, “Models of representations of current groups,” in: Representations of Lie groups and Lie algebras (Budapest, 1971), Akad. Kiado, Budapest (1985), pp. 121–179.

  8. I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5: Integral Geometry and Representation Theory, Amer. Math. Soc. (1966).

  9. E. Glasner, B. Tsirelson, and B. Weiss, “The automorphism group of the Gaussian measure cannot act pointwise,” Israel J. Math., 148, 305–329 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  10. G. W. Mackey, “Point realization of transformation groups,” Illinois J. Math., 6, 327–335 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. A. Rokhlin, “On the fundamental ideas of measure theory,” Mat. Sb., 25(67), 107–150 (1949).

    MathSciNet  Google Scholar 

  12. G. E. Shilov and Fan Dyk Tin, Integral, Measure, and Derivative on Linear Spaces [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  13. A. V. Skorokhod, Integration in Hilbert Space. Springer-Verlag, Berlin–Heidelberg (1974).

    Book  MATH  Google Scholar 

  14. V. N. Sudakov, “Geometric problems of the theory of infinite-dimensional probability distributions,” Proc. Steklov Inst. Math., 141, 1–178 (1979).

    MathSciNet  Google Scholar 

  15. S. Thomas, “A descriptive view of unitary group representations,” J. Europ. Math. Soc., 017.7, 1761–1787 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. M. Vershik, “A measurable realization of continuous groups of automorphisms of a unitary ring,” Izv. Akad. Nauk SSSR, Ser. Mat., 29, 127–136 (1965).

    MathSciNet  Google Scholar 

  17. A. M. Vershik, “Duality in the measure theory in linear spaces,” in: International Congress of Mathematicians, Abstracts, Sec. 5, Moscow (1966), p. 39.

  18. A. M. Vershik, “Duality in the theory of measure in linear spaces,” Sov. Math. Dokl., 7, 1210–1214 (1967).

    MATH  Google Scholar 

  19. A. M. Vershik, “The axiomatics of measure theory in linear spaces,” Sov. Math. Dokl., 9, 68–72 (1968).

    MATH  Google Scholar 

  20. A. M. Vershik, “Measurable realizations of automorphism groups and integral representations of positive operators,” Sib. Mat. Zh., 28, 36–43 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. M. Vershik, “Random and universal metric spaces,” in: Fundamental Mathematics Today, S. K. Lando and O. K. Sheinman (eds.), Independent Univ. Moscow, (2003), pp. 54–88.

    Google Scholar 

  22. A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence,” Russian Math. Surveys, 72, 257–333 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. M. Vershik and V. N. Sudakov, “Probability measures in infinite-dimensional spaces,” Zap. Nauchn. Semin. LOMI, 12, 1–28 (1971).

    Google Scholar 

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Authors and Affiliations

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

    A. M. Vershik

  2. Institute for Information Transmission Problems, Moscow, Russia

    A. M. Vershik

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  1. A. M. Vershik
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Correspondence to A. M. Vershik.

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To the memory of V. N. Sudakov

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 457, 2017, pp. 74–100.

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Vershik, A.M. Duality and Free Measures in Vector Spaces, the Spectral Theory of Actions of Non-Locally Compact Groups. J Math Sci 238, 390–405 (2019). https://doi.org/10.1007/s10958-019-04246-5

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  • DOI: https://doi.org/10.1007/s10958-019-04246-5

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