Skip to main content
SpringerLink
Log in

A new upper bound for the complex Grothendieck constant

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

Abstract

Let ϕ denote the real function

$$\varphi (k) = k\smallint _0^{\pi /2} \frac{{cos^2 t}}{{\sqrt {1 - k^2 sin ^2 t} }}dt, - 1 \leqq k \leqq 1$$

and letK CG be the complex Grothendieck constant. It is proved thatK CG ≦8/π(k 0+1), wherek 0 is the (unique) solution to the equationϕ(k)=1/8π(k+1) in the interval [0,1]. One has 8/π(k 0+1) ≈ 1.40491. The previously known upper bound isK CG ≦e 1−y ≈ 1.52621 obtained by Pisier in 1976.

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. P. F. Byrd and M. D. Friedman,Handbook of Elliptic Integrals for Engineers and Scientists, Die Grundlehren Math. Wiss. in Einzeldarst. 67, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

  2. A. M. Davie, private communication (1984).

  3. A. Grothendieck,Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. Sao Paulo8 (1956), 1–79.

    MathSciNet  Google Scholar 

  4. U. Haagerup,The Grothendieck inequality for bilinear forms on C*-algebras, Advances in Math.56 (1985), 93–116.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Kaijser,A note on the Grothendieck constant with an application to harmonic analysis, UUDM Report No. 1973:10, Uppsala University (mimeographed).

  6. J. L. Krivine,Théorème de factorisation dans les espaces réticulés, exp. XXII–XXIII, Seminaire Maurey-Schwartz, 1973–74.

  7. J. J. Krivine,Constantes de Grothendieck et fonctions de type positif sur les sphères, Advances in Math.31 (1979), 16–30.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Lindenstrauss and A. Pelczynski,Absolutely summing operators in l p-spaces and their applications, Studia Math.29 (1968), 275–326.

    MATH  MathSciNet  Google Scholar 

  9. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Ergebnisse der Mathematik und ihre Grenzgeb.97, Springer-Verlag, Berlin, 1979.

    MATH  Google Scholar 

  10. L. M. Milne-Thomson,Ten-figure tables of the complete elliptic integrals K, K’, E, E’, Proc. London Math. Soc. (2)33 (1932), 160–164.

    Article  Google Scholar 

  11. G. Pisier,Grothendieck’s theorem for non-commutative C*-algebras with an appendix on Grothendieck’s constant, J. Funct. Anal.29 (1978), 379–415.

    Article  MathSciNet  Google Scholar 

  12. R. Rietz,A proof of the Grothendieck inequality, Isr. J. Math.19 (1974), 271–276.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisk Institut, Odense University, DK-5230, Odense M, Denmark

    Uffe Haagerup

Authors
  1. Uffe Haagerup
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Haagerup, U. A new upper bound for the complex Grothendieck constant. Israel J. Math. 60, 199–224 (1987). https://doi.org/10.1007/BF02790792

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02790792

Keywords

Search

Navigation