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Uncomplementability of spaces of compact operators in larger spaces of operators

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Abstract

In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of c 0 in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results in the paper.

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References

  1. J. Bourgain, F. Delbaen: A class of special L ∞ spaces. Acta Math. 145 (1980), 155-176.

    Google Scholar 

  2. R. D. Bourgin: Geometric aspects of convex sets with the Radon-Nikodym property LNM 993. Springer Verlag, 1983.

  3. J. Diestel, J. J. Uhl, jr.: Vector Measures. Math. Surveys 15, Amer. Math. Soc., 1977.

  4. N. Dunford, J. T. Schwartz: Linear Operators, part I. Interscience, 1958.

  5. G. Emmanuele: About certain isomorphic properties of Banach spaces in projective tensor products. Extracta Math. 5(1) (1990), 23-25.

    Google Scholar 

  6. G. Emmanuele: Remarks on the uncomplemented subspace W(E, F). J. Funct. Analysis 99(1) (1991), 125-130.

    Google Scholar 

  7. G. Emmanuele: A remark on the containment of c 0 in spaces of compact operators. Math. Proc. Cambridge Phil. Soc. 111 (1992), 331-335.

    Google Scholar 

  8. G. Emmanuele: About the position of K w*(X*, Y). Atti Seminario Matematico e Fisico di Modena, XLII (1994), 123-133.

    Google Scholar 

  9. G. Emmanuele: Answer to a question by M. Feder about K(X, Y). Revista Mat. Universidad Complutense Madrid 6 (1993), 263-266.

    Google Scholar 

  10. M. Feder: On subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math. 24 (1980), 196-205.

    Google Scholar 

  11. M. Feder: On the non-existence of a projection onto the spaces of compact operators. Canad. Math. Bull. 25 (1982), 78-81.

    Google Scholar 

  12. G. Godefroy, N. J. Kalton, P. D. Saphar: Unconditional ideals in Banach spaces. Studia Math. 104(1) (1993), 13-59.

    Google Scholar 

  13. K. John: On the uncomplemented subspace K(X, Y). Czechoslovak Math. Journal 42 (1992), 167-173.

    Google Scholar 

  14. K. John: On the space K(P, P*) of compact operators on Pisier space P. Note di Matematica 72 (1992), 69-75.

    Google Scholar 

  15. J. Johnson: Remarks on Banach spaces of compact operators. J. Funct. Analysis 32 (1979), 304-311.

    Google Scholar 

  16. W. B. Johnson, H. P. Rosenthal, M. Zippin: On bases, finite dimensional decompositions and weaker structures in Banach spaces. Israel J. Math. 9 (1971), 488-506.

    Google Scholar 

  17. N. J. Kalton: Spaces of compact operators. Math. Annalen 208 (1974), 267-278.

    Google Scholar 

  18. N. J. Kalton: M-ideals of compact operators. Illinois J. Math. 37(1) (1993), 147-169.

    Google Scholar 

  19. D. R. Lewis: Conditional weak compactness in certain inductive tensor products. Math. Annalen 201 (1973), 201-209.

    Google Scholar 

  20. Å. Lima: Uniqueness of Hahn-Banach extensions and lifting of linear dependence. Math. Scandinavica 53 (1983), 97-113.

    Google Scholar 

  21. Å. Lima: The metric approximation property, norm one projections and intersection properties of balls. Israel J. Math. To appear.

  22. Å. Lima, E. Oja, T. S. S. R. K. Rao, D. Werner: Geometry of operator spaces. Preprint 1993.

  23. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces, Sequence Spaces EMG 92. Springer Verlag, 1977.

  24. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces, Function Spaces EMG 97. Springer Verlag, 1979.

  25. I. Namioka, R. R. Phelps: Banach spaces which are Asplund spaces. Duke Math. J. 42 (1975), 735-750.

    Google Scholar 

  26. A. Pelczynski: Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Polon. Sci. 10 (1962), 641-648.

    Google Scholar 

  27. W. Ruess: Duality and Geometry of spaces of compact operators. Functional Analysis: Surveys and Recent Results III, Math. Studies 90. North Holland, 1984.

  28. G. Willis: The compact approximation property does not imply the approximation property. Studia Math. 103(1) (1992), 99-108.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Catania, Viale A. Doria 6, 95125, Catania, Italy

    G. Emmanuele

  2. Mathematical Institute, Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    K. John

Authors
  1. G. Emmanuele
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  2. K. John
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Emmanuele, G., John, K. Uncomplementability of spaces of compact operators in larger spaces of operators. Czechoslovak Mathematical Journal 47, 19–32 (1997). https://doi.org/10.1023/A:1022483919972

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