Advertisement

Geometric & Functional Analysis GAFA

, Volume 5, Issue 2, pp 329–363 | Cite as

Bilinear forms on exact operator spaces andB(H)B(H)

  • M. Junge
  • G. Pisier
Article

Abstract

LetE, F be exact operator spaces (for example subspaces of theC*-algebraK(H) of all the compact operators on an infinite dimensional Hilbert spaceH). We study a class of bounded linear mapsu: EF* which we call tracially bounded. In particular, we prove that every completely bounded (in shortc.b.) mapu: EF* factors boundedly through a Hilbert space. This is used to show that the setOSn of alln-dimensional operator spaces equipped with thec.b. version of the Banach Mazur distance is not separable ifn>2.

As an application we whow that there is more than oneC*-norm onB (H) ⊗ B (H), or equivalently that
$$B(H) \otimes _{\min } B(H) \ne B(H) \otimes _{\max } B(H),$$
which answers a long standing open question. Finally we show that every “maximal” operator space (in the sense of Blecher-Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the “exactness constant”. In the final section, we introduce and study a new tensor product forC*-albegras and for operator spaces, closely related to the preceding results.

Keywords

Hilbert Space Tensor Product Operator Space Final Section Bilinear Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AO]C. Akemann, P. Ostrand, Computing norms in group,C *-algebras, Amer. J. Math. 98 (1976), 1015–1047.Google Scholar
  2. [B]R. Baire, Sur les fonctions des variables réelles, Ann. di Mat. 3∶3 (1899), 1–123.Google Scholar
  3. [Bl1]D. Blecher, Tensor products of operator spaces II, Canadian J. Math. 44 (1992), 75–90.Google Scholar
  4. [Bl2]D. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), 15–30.Google Scholar
  5. [Bl3]D. Blecher, Tracially completely bounded multilinear maps onC *-algebras, Journal of the London Mathematical Society 39 (1989), 514–524.Google Scholar
  6. [BlP]D. Blecher, V. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991) 262–292.Google Scholar
  7. [dCH]J. de Cannière, U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), 455–500.Google Scholar
  8. [dHaV]P. de la Harpe, A. Valette, La Propriété T de Kazhdan pour les Groupes Localement Compacts, Astérisque, Soc. Math. France 175 (1989).Google Scholar
  9. [EH]E. Effros, U. Haagerup, Lifting problems and local reflexivity forC *-algebras, Duke Math. J. 52 (1985), 103–128.Google Scholar
  10. [ER1]E. Effros, Z. J. Ruan, A new approach to operator spaces, Canadian Math. Bull. 34 (1991), 329–337.Google Scholar
  11. [ER2]E. Effros, Z.J. Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), 579–584.Google Scholar
  12. [H1]U. Haagerup, The Grothendieck inequality for bilinear forms onC *-algebras, Advances in Math. 56 (1985), 93–116.Google Scholar
  13. [H2]U. Haagerup, An example of a non-nuclearC *-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279–293.Google Scholar
  14. [H3]U. Haagerup, Injectivity and decomposition of completely bounded maps, in “Operator algebras and their connection with Topology and Ergodic Theory”, Springer Lecture Notes in Math. 1132 (1985), 170–222.Google Scholar
  15. [HPi]U. Haagerup, G. Pisier, Bounded linear operators betweenC *-algebras, Duke Math. J. 71 (1993), 889–925.Google Scholar
  16. [I]T. Itoh, On the completely bounded maps of aC *-algebra to its dual space, Bull. London Math. Soc. 19 (1987), 546–550.Google Scholar
  17. [K1]E. Kirchberg, On subalgebras of the CAR-algebra, to appear in J. Funct. Anal.Google Scholar
  18. [K2]E. Kirchberg, On non-semisplit extensions, tensor products and exactness of groupC *-algebras, Invent. Math. 112 (1993), 449–489.Google Scholar
  19. [K3]E. Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exactness, to appear in J. reine angew. Math.Google Scholar
  20. [Kr]J. Kraus, The slice map problem and approximation properties, J. Funct. Anal. 102 (1991), 116–155.Google Scholar
  21. [Ku]W. Kuratowski, Topology, Vol. 1. (New edition translated from the French), Academic Press, New-York 1966.Google Scholar
  22. [Kw]S. Kwapień, On operators factorizable throughL p-spaces, Bull. Soc. Math. France, Mémoire 31–32 (1972), 215–225.Google Scholar
  23. [L]C. Lance, On nuclearC *-algebras, J. Funct. Anal. 12 (1973), 157–176.Google Scholar
  24. [P1]V. Paulsen, Completely bounded maps and dilations, Pitman Research Notes 146. Pitman Longman (Wiley) 1986.Google Scholar
  25. [P2]V. Paulsen, Representation of function algebras, Abstract operator spaces and Banach space geometry, J. Funct. Anal. 109 (1992), 113–129.Google Scholar
  26. [P3]V. Paulsen, The maximal operator space of a normed space, to appear.Google Scholar
  27. [Pi1]G. Pisier, Exact operator spaces, Colloque sur les algèbres d'opérateurs, Astérisque, Soc. Math. France, to appear.Google Scholar
  28. [Pi2]G. Pisier, Factorization of Linear Operators and the Geometry of Banach Spaces, CBMS (Regional conferences of the A.M.S.) 60, (1986); Reprinted with corrections 1987.Google Scholar
  29. [Pi3]G. Pisier, The operator Hilbert spaceOH, complex interpolation and tensor norms, submitted to Memoirs Amer. Math. Soc.Google Scholar
  30. [Pi4]G. Pisier, Factorization of operator valued analytic functions, Advances in Math. 93 (1992), 61–125.Google Scholar
  31. [Pi5]G. Pisier, Projections from a von Neumann algebra onto a subalgebra, Bull. Soc. Math. France, to appear.Google Scholar
  32. [R]Z. J. Ruan, Subspaces ofC *-algebras, J. Funct. Anal. 76 (1988), 217–230.Google Scholar
  33. [S]S. Sakai,C *-algebras andW *-algebras, Springer Verlag New-York, 1971.Google Scholar
  34. [Sm]R.R. Smith, Completely bounded maps betweenC *-algebras, J. London Math. Soc. 27 (1983), 157–166.Google Scholar
  35. [T]M. Takesaki, Theory of Operator Algebras I, Springer-Verlag New-York 1979.Google Scholar
  36. [Tr]S. Trott, A pair of generators for the unimodular group, Canad. Math. Bull. 3 (1962), 245–252.Google Scholar
  37. [Vo]D. Voiculescu, Property T and approximation of operators, Bull. London Math. Soc. 22 (1990), 25–30.Google Scholar
  38. [VoDN]D. Voiculescu, K. Dykema, A. Nica, Free random variables, CRM Monograph Series, 1, Amer. Math. Soc., Providence RI.Google Scholar
  39. [W1]S. Wassermann, On tensor products of certain groupC *-algebras, J. Funct. Anal. 23 (1976), 239–254.Google Scholar
  40. [W2]S. Wassermann, ExactC *-algebras and related topics, Lecture Notes Series 19, Seoul National University, 1994.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • M. Junge
    • 1
  • G. Pisier
    • 2
    • 3
  1. 1.Mathematisches Seminar CAUKielGermany
  2. 2.Texas A&M UniversityCollege StationUSA
  3. 3.Université Paris 6 Equipe d'AnalyseParis Cedex 05France

Personalised recommendations