Israel Journal of Mathematics

, Volume 60, Issue 2, pp 199–224 | Cite as

A new upper bound for the complex Grothendieck constant

  • Uffe Haagerup


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 G C be the complex Grothendieck constant. It is proved thatK G C ≦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 G C e 1−y ≈ 1.52621 obtained by Pisier in 1976.


Hilbert Space Half Plane Banach Lattice Elliptic Integral Complete Elliptic Integral 
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  1. 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.MATHGoogle Scholar
  2. 2.
    A. M. Davie, private communication (1984).Google Scholar
  3. 3.
    A. Grothendieck,Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. Sao Paulo8 (1956), 1–79.MathSciNetGoogle Scholar
  4. 4.
    U. Haagerup,The Grothendieck inequality for bilinear forms on C*-algebras, Advances in Math.56 (1985), 93–116.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    S. Kaijser,A note on the Grothendieck constant with an application to harmonic analysis, UUDM Report No. 1973:10, Uppsala University (mimeographed).Google Scholar
  6. 6.
    J. L. Krivine,Théorème de factorisation dans les espaces réticulés, exp. XXII–XXIII, Seminaire Maurey-Schwartz, 1973–74.Google Scholar
  7. 7.
    J. J. Krivine,Constantes de Grothendieck et fonctions de type positif sur les sphères, Advances in Math.31 (1979), 16–30.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Lindenstrauss and A. Pelczynski,Absolutely summing operators in l p-spaces and their applications, Studia Math.29 (1968), 275–326.MATHMathSciNetGoogle Scholar
  9. 9.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Ergebnisse der Mathematik und ihre Grenzgeb.97, Springer-Verlag, Berlin, 1979.MATHGoogle Scholar
  10. 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.CrossRefGoogle Scholar
  11. 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.CrossRefMathSciNetGoogle Scholar
  12. 12.
    R. Rietz,A proof of the Grothendieck inequality, Isr. J. Math.19 (1974), 271–276.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1987

Authors and Affiliations

  • Uffe Haagerup
    • 1
  1. 1.Mathematisk InstitutOdense UniversityOdense MDenmark

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