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On the Duality of Operator Spaces

Published online by Cambridge University Press:  20 November 2018

Christian Le Merdy
Christian Le Merdy*
Affiliation:
Equipe de Mathématiques, URA CNRS 741, Université de Franche-Comté, F-25030 Besançon Cedex, France
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Abstract

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We prove that given an operator space structure on a dual Banach space Y*, it is not necessarily the dual one of some operator space structure on Y. This allows us to show that Sakai's theorem providing the identification between C*-algebras having a predual and von Neumann algebras does not extend to the category of operator spaces. We also include a related result about completely bounded operators from B(ℓ2)* into the operator Hilbert space OH.

Keywords

47C1546A2046B28
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 3 , 01 September 1995 , pp. 334 - 346
Copyright
Copyright © Canadian Mathematical Society 1995

References

[AO] Akeman, C. and Ostrand, P., Computing norms in group C* -algebras, Amer. J. Math. 98(1976), 1015— 1047.Google Scholar
[B1] Blecher, D., Tensor products of operator spaces II, Canad. J. Math. 44(1992), 75–90.Google Scholar
[B2] Blecher, D., The standard dual of an operator space, Pacific J. Math. 153(1992), 15–30.Google Scholar
[BP] Blecher, D. and Paulsen, V., Tensor products of operator spaces, J. Funct. Anal. 99(1991), 262—292.Google Scholar
[BS] Blecher, D. and Smith, R., The dual of the Haagerup tensor product, J. London Math. Soc. 45(1992), 126–144.Google Scholar
[D] Dean, D.. The equation L(E,X**) = L(E,X)** and the principle of local reflexivity, Proc. Amer. Math. Soc. 40(1973), 146–148.Google Scholar
[EH] Effros, E. and Haagerup, U., Lifting problems and local reflexivity for C*-algebras, Duke Math. J. 52 (1985), 103–128.Google Scholar
[ER1] Effros, E. and Ruan, Z.-J., A new approach to operator spaces, Canad. Math. Bull. 34(1991), 329–337.Google Scholar
[ER2] Effros, E., Self duality for the Haagerup tensor product and Hilbert space factorization, J. Func. Anal. 100(1991), 257–284.Google Scholar
[ER3] Effros, E., Mapping spaces and liftings for operator spaces, Proc. London Math. Soc. 69( 1994), 171—197.Google Scholar
[ER4] Effros, E.,. On approximation properties for operator spaces, Internat. J. Math. 1(1990), 163—187.Google Scholar
[HP] Haagerup, U. and Pisier, G., Bounded linear operators between C*-algebras, Duke Math. J. 71(1993), 889–925.Google Scholar
[L] Le Merdy, C., Analytic factorizations and completely bounded maps, Israel J. Math. 88(1994), 381–409.Google Scholar
[P1] Pisier, G., The operator Hilbert space OH, complex interpolation and tensor norms, to appear.Google Scholar
[P2] Pisier, G., Espace de Hilbert d'opérateurs et interpolation complexe, C. R. Acad. Sci. Paris Série I Math. 316(1993), 47–52.Google Scholar
[P3] Pisier, G., Factorization of linear operators and geometry ofBanach spaces, CBMS Regional Conf. Amer. Math. Soc. 60(1986).Google Scholar
[R] Ruan, Z.-J.. Subspaces of C*-algebras, J. Funct. Anal. 76(1988), 217–230.Google Scholar
[S1] Sakai, S.. A characterization of W*-algebras, Pacific J. Math. 6(1956), 763–773.Google Scholar
[S2] Sakai, S., C*-algebras and W*-algebras, Springer Verlag, 1971.Google Scholar
[T] Takesaki, M., Theory of operator algebras I, Springer Verlag, 1979.Google Scholar