Mathematische Annalen

, Volume 347, Issue 2, pp 299–338 | Cite as

Maurey’s factorization theory for operator spaces

Article

Abstract

We prove an operator space version of Maurey’s theorem, which claims that every absolutely (p, 1)-summing map on C(K) is automatically absolutely q-summing for q > p. Our results imply in particular that every completely bounded map from B(H) with values in Pisier’s operator space OH is completely p-summing for p > 2. This fails for p = 2. As applications, we obtain eigenvalue estimates for translation invariant maps defined on the von Neumann algebra VN(G) associated with a discrete group G. We also develop a notion of cotype which is compatible with factorization results on noncommutative Lp spaces.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bergh J., Löfström J.: Interpolation Spaces. Springer, Berlin (1976)MATHGoogle Scholar
  2. 2.
    Choi M.D., Effros E.G.: The completely positive lifting problem for C *-algebras. Ann. Math. 104, 585–609 (1976)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Effros E.G., Ruan Z.J.: A new approach to operator spaces. Can. Math. Bull. 34, 329–337 (1991)MATHMathSciNetGoogle Scholar
  4. 4.
    Effros, E.G., Ruan, Z.J.: Operator Spaces. London Math. Soc. Monogr. 23, Oxford University Press, New York (2000)Google Scholar
  5. 5.
    García-Cuerva J., Parcet J.: Quantized orthonormal systems: a non-commutative Kwapień theorem. Studia Math. 155, 273–294 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Haagerup, U.: Self-polar forms, conditional expectations and the weak expectation property for C *-algebras. Unpublished manuscript (1995)Google Scholar
  7. 7.
    Haagerup, U., Junge, M., Xu, Q.: A reduction method for noncommutative L p spaces and applications. Trans. Amer. Math. Soc. (2009)Google Scholar
  8. 8.
    Haagerup U., Musat M.: The Effros-Ruan conjecture for bilinear forms on C *-algebras. Invent. Math. 174, 139–163 (2008)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Junge, M.: Factorization Theory for Spaces of Operators. Habilitation Thesis. Kiel (1996)Google Scholar
  10. 10.
    Junge M.: Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549, 149–190 (2002)MATHMathSciNetGoogle Scholar
  11. 11.
    Junge M.: Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’. Invent. Math. 161, 225–286 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Junge, M.: Vector-valued L p spaces over QWEP von Neumann algebras. PreprintGoogle Scholar
  13. 13.
    Junge M., Lee H.H.: A Maurey type result for operator spaces. J. Funct. Anal. 254, 1373–1409 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Junge M., Parcet J.: The norm of sums of independent noncommutative random variables in L p( 1). J. Funct. Anal. 221, 366–406 (2005)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Junge M., Parcet J.: Rosenthal’s theorem for subspaces of noncommutative L p. Duke Math. J. 141, 75–122 (2008)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Junge M., Parcet J.: Operator space embedding of Schatten p-classes into von Neumann algebra preduals. Geom. Funct. Anal. 18, 522–551 (2008)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Junge, M., Parcet, J.: Mixed-norm inequalities and operator space L p embedding theory. Mem. Amer. Math. Soc (to appear)Google Scholar
  18. 18.
    Junge, M., Parcet, J.: A transference method in quantum probability. PreprintGoogle Scholar
  19. 19.
    Junge M., Sherman D.: Noncommutative L p modules. J. Operator Theory 53, 3–34 (2005)MATHMathSciNetGoogle Scholar
  20. 20.
    Junge M., Xu Q.: Noncommutative Burkholder/Rosenthal inequalities II: applications. Israel J. Math. 167, 227–282 (2008)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Junge, M., Xu, Q.: Representation of certain homogeneous Hilbertian operator spaces and applications. PreprintGoogle Scholar
  22. 22.
    Junge, M., Xu, Q.: Interpolation for noncommutative L p(L q) spaces and applications. In progressGoogle Scholar
  23. 23.
    Kirchberg E.: On nonsemisplit extensions, tensor products and exactness of group C *-algebras. Invent. Math. 112, 449–489 (1993)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kosaki H.: Applications of the complex interpolation method to a von Neumann algebra. J. Funct. Anal. 56, 29–78 (1984)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Krivine J.L.: Sous-espaces de dimension finie des espaces de Banach réticulé. Ann. Math. 104, 1–29 (1976)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Lee H.H.: Type and cotype of operator spaces. Studia Math. 185, 219–247 (2008)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lindenstrauss J., Pełczyński A.: Absolutely summing operators in L p-spaces and their applications. Studia Math. 29, 275–326 (1968)MATHMathSciNetGoogle Scholar
  28. 28.
    Maurey, B.: Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces L p. Asterisque 11 (1974)Google Scholar
  29. 29.
    Maurey B., Pisier G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studia Math. 58, 45–90 (1976)MATHMathSciNetGoogle Scholar
  30. 30.
    Parcet, J.: Análisis armónico no conmutativo y geometría de espacios de operadores. Ph.D. Thesis, (2003)Google Scholar
  31. 31.
    Parcet J.: B-convex operator spaces. Proc. Edinburgh Math. Soc. 46, 649–668 (2003)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Parcet J., Pisier G.: Non-commutative Khintchine type inequalities associated with free groups. Indiana Univ. Math. J. 54, 531–556 (2005)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Pietsch, A.: Eigenvalues and s-numbers. In: Cambridge Studies in Advanced Mathematics, vol. 13. Cambridge University Press, Cambridge (1987)Google Scholar
  34. 34.
    Pisier G.: Factorization of operators through L p or L p1 and noncommutative generalizations. Math. Ann. 276, 105–136 (1986)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Pisier G.: Projections from a von Neumann algebra onto a subalgebra. Bull. Soc. Math. France 123, 139–153 (1995)MATHMathSciNetGoogle Scholar
  36. 36.
    Pisier G.: Dvoretzky’s theorem for operator spaces. Houston J. Math. 22, 399–416 (1996)MATHMathSciNetGoogle Scholar
  37. 37.
    Pisier, G.: The Operator Hilbert space OH, complex interpolation and tensor norms. Mem. Am. Math. Soc. 122 (1996)Google Scholar
  38. 38.
    Pisier, G.: Non-commutative vector valued L p-spaces and completely p-summing maps. Astérisque 247 (1998)Google Scholar
  39. 39.
    Pisier G.: Introduction to operator space theory. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  40. 40.
    Pisier G., Shlyakhtenko D.: Grothendieck’s theorem for operator spaces. Invent. Math. 150, 185–217 (2002)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Raynaud Y.: On ultrapowers of non-commutative L p spaces. J. Operator Theory 48, 41–68 (2002)MATHMathSciNetGoogle Scholar
  42. 42.
    Rosenthal H.P.: On subspaces of L p. Ann. Math. 97, 344–373 (1973)CrossRefGoogle Scholar
  43. 43.
    Xu Q.: A description of (C p[L p(M)], R p[L p(M)])θ. Proc. Roy. Soc. Edinburgh Sect. A 135, 1073–1083 (2005)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Xu Q.: Operator-space Grothendieck inequalities for noncommutative L p-spaces. Duke Math. J. 131, 525–574 (2006)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Xu Q.: Embedding of C q and R q into noncommutative L p-spaces, 1 ≤ p < q ≤ 2. Math. Ann. 335, 109–131 (2006)MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Xu, Q.: Real interpolation approach to Junge’s works on embedding of OH and the little Grothendieck inequality. PreprintGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Instituto de Ciencias MatemáticasCSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones CientíficasMadridSpain

Personalised recommendations