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Negative Definite Functions on the Space of Infinite Hermitian Matrices

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Abstract

In this paper we investigate a Lévy–Khinchin type integral formula of negative definite functions defined over the Olshanski spherical pair of infinite hermitian matrices.

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References

  1. Berg, C., Christensen, C., Ressel, J.P.R.: Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions. Springer, New York (1984)

    Book  Google Scholar 

  2. Berg, C., Forst, G.: Potentiel Theory on Locally Compact Abelian Groups. Springer-Verlag, Berlin, Heidelberg (1975)

    Book  Google Scholar 

  3. Bouali, M.: A Lévy–Khinchin formula for the space of infinite dimensional hermitian matrices. J. Lie Theory 18(1), 017–032 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Bouali, M.: Lévy–Khinchin formula for the infinite symmetric group. Mathematische Zeitschrift 273(1–2), 303–310 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bouali, M.: Application des théorèmes de Minlos et Poincaré à l’étude asymptotique d’une intégrale orbitale. Ann. Fac. Sci. Toulouse Math. (6) 16, 49–70 (2007)

    Article  MathSciNet  Google Scholar 

  6. Bouali, M.: Analyse harmonique en dimension infinie. Thèse de doctorat de l’université Pierre et Marie Curie - Paris VI (2006)

  7. Borodin, A., Olshanski, G.: Infinite random matrices and ergodic measures. Commun. Math. Phys. 223, 87–123 (2001)

    Article  MathSciNet  Google Scholar 

  8. Faraut, J.: Asymptotic spherical analysis on the Heisenberg group. Colloq. Math. 118, 233–258 (2010)

    Article  MathSciNet  Google Scholar 

  9. Faraut, J.: Olshanski spherical pairs related to the Heisenberg group. Mosc. Math. J. 14, 63–81 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Faraut, J.: Infinite dimensional harmonic analysis and probability. In: Dani, S.G., Graczyk, P. (eds.) Probability Measures on Groups: Recent Directions and Trends, pp. 179–254. Tata Institute of Fundamental Research, Narosa (2006)

    MATH  Google Scholar 

  11. Olshanski, G., Vershik, A.: Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. Am. Math. Soc. Transl. 2(175), 137–175 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Olshanski, G., Vershik, A.: The problem of harmonic analysis on the infinite dimensional unitary group. J. Funct. Anal. 205, 464–524 (2003)

    Article  MathSciNet  Google Scholar 

  13. Olshanski, G., Borodin, A.: Infinite random matrices and ergodic measures. Commun. Math. Phys. 223(1), 87–123 (2001)

    Article  MathSciNet  Google Scholar 

  14. Olshanski, G., Borodin, A.: Unitary representations of infinite dimensional pairs (G,K) and the formalism of R. Howe. In: Vershik, A.M., Zhelobenko, D.P. (eds.) Representations of Lie groups and Related Topics. Advanced Studies in Contemporary Mathematics, vol. 7, pp. 269463. Gordon and Breach Science Publishers, New York (1990)

  15. Rabaoui, M.: A Bochner type theorem for inductive limits of Gelfand pairs. Annales de l’institut Fourier 58(5), 1551–1573 (2008)

    Article  MathSciNet  Google Scholar 

  16. Rabaoui, M.: A Lévy–Khinchin formula for the space of infinite dimensional Square Complex Matrices. Bull. Sci. Math. 139, 283–300 (2015)

    Article  MathSciNet  Google Scholar 

  17. Rabaoui, M.: On functions of negative type on the Olshanski spherial pair \((SL(\infty ), SU(\infty ))\). J. Lie Theory 2(1), 237–250 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. Math. Second Ser. 39(4), 811–841 (1938)

    Article  MathSciNet  Google Scholar 

  19. Thoma, E.: Die unzerlegbern, positive-definite klassenfunctionen der abzahlbar unendlichen, symmetrischen gruppe. Math. Z. 85, 40–61 (1964)

    Article  MathSciNet  Google Scholar 

  20. Voiculescu, D.: Repésentation factorielles de type \(II_1\) de \(U(1)\). J. Math. pures et appl 55, 1–22 (1976)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Mohamed Bouali.

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Communicated by Daniel Aron Alpay.

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Bouali, M. Negative Definite Functions on the Space of Infinite Hermitian Matrices. Complex Anal. Oper. Theory 12, 1707–1727 (2018). https://doi.org/10.1007/s11785-018-0791-8

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  • DOI: https://doi.org/10.1007/s11785-018-0791-8

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