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On p-Convexity and q-Concavity in Non-Commutative Symmetric Spaces

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Abstract

It is shown that if a symmetric Banach space E on the positive semi-axis is p-convex (q-concave) then so is the corresponding non-commutative symmetric space E(τ) of τ-measurable operators affiliated with some semifinite von Neumann algebra \({({\mathcal{M}}, \tau)}\) , with preservation of the convexity (concavity) constants in the case that \({{\mathcal{M}}}\) is non-atomic. Similar statements hold in the case that E satisfies an upper (lower) p-estimate and extend to the more general semifinite setting earlier results due to Arazy and Lin for unitary matrix spaces.

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References

  1. Aliprantis, C.D., Burkinshaw O.: Locally solid Riesz spaces with applications to economics. In: Mathematics Surveys and Monographs vol. 105, 2nd edn., American mathematical society (2003)

  2. Arazy J., Lin P-K.: On p-convexity and q-concavity of unitary matrix spaces. Int. Equ. Oper. Theory 8, 295–313 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chilin V.I., Sukochev F.A.: Weak convergence in non-commutative symmetric spaces. J. Oper. Theory 31, 35–65 (1994)

    MATH  MathSciNet  Google Scholar 

  4. Chilin V.I., Krygin A.V., Sukochev F.A.: Local uniform and uniform convexity of non-commutive symmetric spaces of measurable operators. Math. Proc. Camb. Phil. Soc. 111, 355–368 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dirksen, S.: Non-Commutative and Vector-Valued Rosenthal Inequalities. Thesis, Delft University of Technology (2011)

  6. Dixmier J.: Von Neumann Algebras, Mathematical Library, vol 27. Amsterdam, North Holland (1981)

    Google Scholar 

  7. Dodds P.G.: Indices for Banach lattices. Indag. Math. 80, 73–86 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dodds P.G., Dodds T.K.: Non-commutative Banach function spaces. Math. Z. 201, 583–597 (1995)

    Article  MathSciNet  Google Scholar 

  9. Dodds P.G., Dodds T.K.-Y., de Pagter B.: Fully symmetric operator spaces. Integr. Equ. Oper. Th. 15, 942–972 (1992)

    Article  MATH  Google Scholar 

  10. Dodds P.G., Dodds T.K.-Y., de Pagter B.: Noncommutative Köthe duality. Trans. Am. Math. Soc. 339, 717–750 (1993)

    MATH  Google Scholar 

  11. Dodds, P.G., de Pagter, B.: Normed Köthe spaces: a non-commutative viewpoint. Indagationes Mathematicae. doi:10.1016/j.indag.2013.01.009 (in press) (2013)

  12. Fack T., Kosaki H.: Generalized s-numbers of τ-measurable operators. Pac. J. Math. 123, 269–300 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grobler J.J.: Indices for Banach function spaces. Math. Z. 145, 99–109 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kalton, N., Montgomery-Smith, S.: Interpolation of Banach spaces. In: Handbook of the geometry of Banach spaces, vol. 2. pp. 1131–1175, North-Holland, Amsterdam (2003)

  15. Kalton N., Sukochev F.: Symmetric norms and spaces of operators. J. Reine Angew. Math. 621, 81–121 (2008)

    MATH  MathSciNet  Google Scholar 

  16. Kreĭn, S.G., Petunīn, Y.Ī., Semënov, E.M.: Interpolation of linear operators. In: Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence (1982)

  17. Krygin A.V.: p-convexity snd q-concavity of non-commutative symmetric spaces (Russian). Dokl. Akad. Nauk. UzSSR 2, 7–8 (1990)

    MathSciNet  Google Scholar 

  18. Lord, S., Sukochev, F., Zanin, D.: Singular traces: theory and applications. In: De Gruyter Studies in Mathematics, vol. 46 (2013)

  19. Lotz H.P.: Rearrangement invariant continuous linear functionals on weak L 1. Positivity 12, 119–132 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lindenstrauss J., Tzafriri L.: Classical Banach Spaces II. Springer-Verlag, Berlin (1979)

    Book  MATH  Google Scholar 

  21. Marshall A.W., Olkin I.: Inequalities: Theory of Majorisation and Its Applications. Academic Press, USA (1979)

    Google Scholar 

  22. von Neumann J.: Einige Sätze über messbare Abbildungen. Ann. Math. 33, 574–586 (1932)

    Article  MATH  MathSciNet  Google Scholar 

  23. Okada, S., Ricker, W.J., Sànchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. In: Operator Theory, Advances and Applications vol. 180. Birkhäuser (2008)

  24. de Pagter B.: Non-commutative Banach function spaces. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity, Trends in Mathematics, pp. 197–227. Birkhäuser Verlag, Basel (2007)

    Google Scholar 

  25. Shimogaki T.: Exponents of norms in semi-ordered linear spaces. Bull. Acad. Polon. Sci. Ser. Math. Astr. et Phys. 13, 135–140 (1965)

    MATH  MathSciNet  Google Scholar 

  26. Strătilă S., Zsidó L.: Lectures on von Neumann algebras. Editura Academiei, Bucharest (1979)

  27. Sukochev, F.A.: Completeness of quasi-normed symmetric operator spaces. In: Indagationes Math., Zaanen Centennial volume, (to appear) (2013)

  28. Xu Q.: Analytic functions with values in lattices and symmetric spaces of measurable operators. Math. Proc. Camb. Phil. Soc. 109, 541–563 (1991)

    Article  MATH  Google Scholar 

  29. Zaanen A.C.: Riesz Spaces II. Amsterdam, North-Holland (1983)

    MATH  Google Scholar 

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Correspondence to P. G. Dodds.

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This work was partially supported by the Australian Research Council.

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Dodds, P.G., Dodds, T.K. & Sukochev, F.A. On p-Convexity and q-Concavity in Non-Commutative Symmetric Spaces. Integr. Equ. Oper. Theory 78, 91–114 (2014). https://doi.org/10.1007/s00020-013-2082-0

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