Skip to main content
SpringerLink
Log in

Factorization in SL∞

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We show that the non-separable Banach space SL∞ is primary. This is achieved by directly solving the infinite-dimensional factorization problem in SL∞. In particular, we bypass Bourgain’s localization method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Alspach, P. Enflo and E. Odell, On the structure of separable Lp spaces (1 < p < ∞), Studia Mathematica 60 (1977), 79–90.

    MathSciNet  MATH  Google Scholar 

  2. A. D. Andrew, Perturbations of Schauder bases in the spaces C(K) and Lp, p > 1, Studia Mathematica 65 (1979), 287–298.

    MathSciNet  MATH  Google Scholar 

  3. A. Arias and J. D. Farmer, On the structure of tensor products of lp-spaces, Pacific Journal of Mathematics 175 (1996), 13–37.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Blower, The Banach space B(l2) is primary, Bulletin of the London Mathematical Society 22 (1990), 176–182.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bourgain, On the primarity of H∞-spaces, Israel Journal of Mathematics 45 (1983), 329–336.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Capon, Primarité de lp(L1), Mathematische Annalen 250 (1980), 55–63.

    MathSciNet  MATH  Google Scholar 

  7. M. Capon, Primarité de Lp(lr), 1 < p, r < ∞, Israel Journal of Mathematics 36 (1980), 346–364.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Capon, Primarité de certains espaces de Banach, Proceedings of the London Mathematical Society 45 (1982), 113–130.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Capon, Primarité de Lp(Lr), 1 < p, r < ∞, Israel Journal of Mathematics 42 (1982), 87–98.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Capon, Primarité de Lp(X), Transactions of the American Mathematical Society 276 (1983), 431–487.

    MathSciNet  MATH  Google Scholar 

  11. P. G. Casazza, C. A. Kottman and B. L. Lin, On some classes of primary Banach spaces, Canadian Journal of Mathematics 29 (1977), 856–873.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. G. Casazza and B. L. Lin, Projections on Banach spaces with symmetric bases, Studia Mathematica 52 (1974), 189–193.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Enflo and T. W. Starbird, Subspaces of L1 containing L1, Studia Mathematica 65 (1979), 203–225.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. L. B. Gamlen and R. J. Gaudet, On subsequences of the Haar system in Lp [1, 1] (1 ≤ p≤∞), Israel Journal of Mathematics 15 (1973), 404–413.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. M. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, Mathematics Lecture Notes Series, W. A. Benjamin, Reading, MA–London–Amsterdam, 1973.

    MATH  Google Scholar 

  16. W. B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Memoirs of the American Mathematical Society 19 (1997).

  17. P. W. Jones, BMO and the Banach space approximation problem, American Journal of Mathematics 107 (1985), 853–893.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. W. Jones and P. F. X. Müller, Conditioned Brownian motion and multipliers into SL ∞, Geometric and Functional Analysis 14 (2004), 319–379.

    Article  MathSciNet  MATH  Google Scholar 

  19. N. J. Laustsen, R. Lechner and P. F. X. Müller, Factorization of the identity through operators with large diagonal, Journal of Functional Analysis (2018), in press, https://doi.org/10.1016/j.jfa.2018.02.010.

    Google Scholar 

  20. R. Lechner, Factorization in mixed norm Hardy and BMO spaces, Studia Mthematica 242 (2018), 231–265.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Lechner and P. F. X. Müller, Localization and projections on bi-parameter BMO, Quarterly Journal of Mathematics 66 (2015), 1069–1101.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Lindenstrauss, On complemented subspaces of m, Israel Journal of Mathematics 5 (1967), 153–156.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer, Berlin–New York, 1977.

  24. B. Maurey, Sous-espaces complémentés de Lp, d’après P. Enflo, in Séminaire Maurey–Schwartz 1974–1975: Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. No. III, Centre deMathématiques, École Polytechnique, Paris, 1975, 18 pp. (erratum, p. 1).

    MATH  Google Scholar 

  25. P. F. X. Müller, On projections in H1 and BMO, Studia Mathematica 89 (1988), 145–158.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. F. X. Müller, Orthogonal projections on martingale H1 spaces of two parameters, Illinois Journal of Mathematics 38 (1994), 554–573.

    MathSciNet  MATH  Google Scholar 

  27. P. F. X. Müller, Isomorphisms between H1 spaces, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series), Vol. 66, Birkhäuser Verlag, Basel, 2005.

  28. P. F. X. Müller, Two remarks on primary spaces, Mathematical Proceedings of the Cambridge Philosophical Society 153 (2012), 505–523.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Pe-lczyński, Projections in certain Banach spaces, Studia Mathematica 19 (1960), 209–228.

    Article  MathSciNet  Google Scholar 

  30. H. M. Wark, A class of primary Banach spaces, Journal of Mathematical Analysis and Applications 326 (2007), 1427–1436.

    Article  MathSciNet  MATH  Google Scholar 

  31. H. M. Wark, The l∞ direct sum of Lp (1 < p < ∞) is primary, Journal of the London Mathematical Society 75 (2007), 176–186.

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, Vol. 25, Cambridge University Press, Cambridge, 1991.

Download references

Author information

Authors and Affiliations

  1. Institute of Analysis, Johannes Kepler University Linz, Altenberger Strasse 69, A-4040, Linz, Austria

    Richard Lechner

Authors
  1. Richard Lechner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard Lechner.

Additional information

Supported by the Austrian Science Foundation (FWF) Pr.Nr. P28352.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lechner, R. Factorization in SL∞. Isr. J. Math. 226, 957–991 (2018). https://doi.org/10.1007/s11856-018-1719-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1719-8

Search

Navigation