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B(H) does not have the approximation property. (English) Zbl 0486.46012

MSC:
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20 Geometry and structure of normed linear spaces
46A32 Spaces of linear operators; topological tensor products; approximation properties
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46M05 Tensor products in functional analysis
47L05 Linear spaces of operators
47L30 Abstract operator algebras on Hilbert spaces
46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
05A17 Combinatorial aspects of partitions of integers
15B57 Hermitian, skew-Hermitian, and related matrices
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References:
[1] Choi, M. &Effros, E., NuclearC *-algebras and the approximation property.Amer. J. Math., 100 (1978), 61–79. · Zbl 0397.46054 · doi:10.2307/2373876
[2] Enflo, P., A counterexample to the approximation property in Banach spaces.Acta Math., 130 (1973), 309–317. · Zbl 0267.46012 · doi:10.1007/BF02392270
[3] Grothendieck, A., Produits tensoriels topologiques et espaces nuclearies.Memoirs Amer. Math. Soc., 16 (1955). · Zbl 0123.30301
[4] Haagerup, U., An example of a non-nuclearC *-algebra, which has the metric approximation property.Invent. Math., 50 (1979), 279–293. · Zbl 0408.46046 · doi:10.1007/BF01410082
[5] Lance, C., On nuclearC *-algebras,J. Functional Analysis, 12 (1973), 157–176. · Zbl 0252.46065 · doi:10.1016/0022-1236(73)90021-9
[6] Lindenstrauss, J. & Tzafriri, L.,Classical Banach Spaces, Vol. 1. Springer-Verlag 1977. · Zbl 0362.46013
[7] Szankowski, A., The space of all bounded operators on Hilbert space does not have the approximation property, exposĂ©, XIV–XV.Seminaire d’analyse fonctionnelle, 1978–79, Ecole Polytechnique.
[8] Takesaki, M., On the cross-norm of the direct product ofC *-algebras.Tohoku Math. J., 16 (1964), 111–122. · Zbl 0127.07302 · doi:10.2748/tmj/1178243737
[9] Wasserman, S., On tensor products of certain groupC *-algebras.J. Functional Analysis, 23 (1976), 239–254. · Zbl 0358.46040 · doi:10.1016/0022-1236(76)90050-1
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