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Uncomplementability of spaces of compact operators in larger spaces of operators. (English) Zbl 0903.46006
Summary: 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.

46A32 Spaces of linear operators; topological tensor products; approximation properties
46B20 Geometry and structure of normed linear spaces
46H10 Ideals and subalgebras
46B99 Normed linear spaces and Banach spaces; Banach lattices
46B25 Classical Banach spaces in the general theory
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[1] J. Bourgain, F. Delbaen: A class of special \({\mathcal L}_\infty \) spaces. Acta Math. 145 (1980), 155-176. · Zbl 0466.46024
[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. · Zbl 0369.46039
[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. · Zbl 1230.46058
[6] G. Emmanuele: Remarks on the uncomplemented subspace \(W(E, F)\). J. Funct. Analysis 99 (1) (1991), 125-130. · Zbl 0769.46006
[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. · Zbl 0774.46019
[8] G. Emmanuele: About the position of \(K_{w^\ast }(X^\ast , Y)\) inside \(L_{w^\ast }(X^\ast , Y)\). Atti Seminario Matematico e Fisico di Modena, XLII (1994), 123-133. · Zbl 0805.46022
[9] G. Emmanuele: Answer to a question by M. Feder about \(K(X, Y)\). Revista Mat. Universidad Complutense Madrid 6 (1993), 263-266. · Zbl 0813.46013
[10] M. Feder: On subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math. 24 (1980), 196-205. · Zbl 0411.46009
[11] M. Feder: On the non-existence of a projection onto the spaces of compact operators. Canad. Math. Bull. 25 (1982), 78-81. · Zbl 0432.46008
[12] G. Godefroy, N. J. Kalton, P. D. Saphar: Unconditional ideals in Banach spaces. Studia Math. 104 (1) (1993), 13-59. · Zbl 0814.46012
[13] K. John: On the uncomplemented subspace \(K(X, Y)\). Czechoslovak Math. Journal 42 (1992), 167-173. · Zbl 0776.46016
[14] K. John: On the space \(K(P, P^\ast )\) of compact operators on Pisier space \(P\). Note di Matematica 72 (1992), 69-75. · Zbl 0802.46015
[15] J. Johnson: Remarks on Banach spaces of compact operators. J. Funct. Analysis 32 (1979), 304-311. · Zbl 0412.47024
[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. · Zbl 0217.16103
[17] N. J. Kalton: Spaces of compact operators. Math. Annalen 208 (1974), 267-278. · Zbl 0266.47038
[18] N. J. Kalton: M-ideals of compact operators. Illinois J. Math. 37 (1) (1993), 147-169. · Zbl 0824.46029
[19] D. R. Lewis: Conditional weak compactness in certain inductive tensor products. Math. Annalen 201 (1973), 201-209. · Zbl 0234.46069
[20] Å. Lima: Uniqueness of Hahn-Banach extensions and lifting of linear dependence. Math. Scandinavica 53 (1983), 97-113. · Zbl 0532.46003
[21] Å. Lima: The metric approximation property, norm one projections and intersection properties of balls. Israel J. Math · Zbl 0814.46016
[22] Å. Lima, E. Oja, T. S. S. R. K. Rao, D. Werner: Geometry of operator spaces. Preprint 1993. · Zbl 0823.46023
[23] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces, Sequence Spaces EMG 92. Springer Verlag, 1977. · Zbl 0362.46013
[24] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces, Function Spaces EMG 97. Springer Verlag, 1979. · Zbl 0403.46022
[25] I. Namioka, R. R. Phelps: Banach spaces which are Asplund spaces. Duke Math. J. 42 (1975), 735-750. · Zbl 0332.46013
[26] A. Pelczynski: Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Polon. Sci. 10 (1962), 641-648. · Zbl 0107.32504
[27] W. Ruess: Duality and Geometry of spaces of compact operators. Functional Analysis: Surveys and Recent Results III, Math. Studies 90, North Holland, 1984. · Zbl 0573.46007
[28] G. Willis: The compact approximation property does not imply the approximation property. Studia Math. 103 (1) (1992), 99-108. · Zbl 0814.46017
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