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HB-subspaces and Godun sets of subspaces in Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

Eve Oja
Eve Oja
Affiliation:
Department of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia.
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Abstract

Let X be a Banach space and Y its closed subspace having property U in X. We use a net (Aα) of continuous linear operators on X such that ‖ Aα ‖ ≤ 1, Aα (X) ⊂ Y for all α, and limαg(Aαy) = g(y), y ∈ Y, g∈Y* to obtain equivalent conditions for Y to be an HB-subspace, u-ideal or h-ideal of X. Some equivalent renormings of c0 and l2 are shown to provide examples of spaces X for which K(X) has property U in L(X) without being an HB-subspace. Considering a generalization of the Godun set [3], we establish some relations between Godun sets of Banach spaces and related operator spaces. This enables us to prove e.g., that if K(X) is an HB-subspace of L(X), then X is an HB-subspace of X**—the result conjectured to be true by Å. Lima [9].

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 120 - 132
Copyright
Copyright © University College London 1997

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References

1.Casazza, P. G. and Kalton, N. J.. Notes on approximation properties in separable Banach spaces. In Geometry of Banach spaces, Proc. Conf. Strobl 1989 (eds. Müller, P. F. X. and Schachermayer, W.). London Math. Soc. Lecture Notes Series no. 158 (Cambridge University Press, 1990), pp. 49–63.Google Scholar
2.Godefroy, G., Kalton, N. J. and Saphar, P. D.. Idéaux inconditionnels dans les espaces de Banach. C.R. Acad. Sci. Paris, 313, Sér. 1 (1991), 845–849.Google Scholar
3.Godefroy, G., Kalton, N. J. and Saphar, P. D.. Unconditional ideals in Banach spaces. Studia Math., 104 (1993), 13–59.CrossRefGoogle Scholar
4.Harmand, P., Werner, D. and Werner, W.. M-ideals in Banach spaces and Banach algebras. Lecture Notes in Math. Vol. 1547 (Springer-Verlag, 1993).Google Scholar
5.Hennefeld, J.. M-ideals, HB-subspaces, and compact operators. Indiana Univ. Math. J., 28 (1979), 927–934.CrossRefGoogle Scholar
6.Johnson, J.. Remarks on Banach spaces of compact operators. J. Fund. Anal., 32 (1979), 304–311.Google Scholar
7.Johnson, J. and Wolfe, J.. On the norm of the canonical projection of E*** onto E1. Proc. Amer. Math. Soc., 75 (1979), 50–52.Google Scholar
8.Lima, A.. On M-ideals and best approximation. Indiana Univ. Math. J., 31 (1982), 27–36.Google Scholar
9.Lima, A.. Uniqueness of Hahn-Banach extensions and liftings of linear dependences. Math. Scand., 53 (1983), 97–113.CrossRefGoogle Scholar
10.Lima, A.. Property (wM*) and the unconditional metric compact approximation property. Studia Math., 113 (1995), 249–263.Google Scholar
11.Lima, A., Oja, E., Rao, T. S. S. R. K. and Werner, D.. Geometry of operator spaces. Michigan Math. J., 41 (1994), 473–490.CrossRefGoogle Scholar
12.Lindenstrauss, J. and Tzafriri, L.. On the complemented subspaces problem. Israel J. Math., 9 (1971), 263–269.Google Scholar
13.Oja, E.. On the uniqueness of the norm preserving extension of a linear functional in the Hahn-Banach theorem. Izv. Akad. Nauk Est. SSR, 33 (1984). 424–438 (in Russian).Google Scholar
14.Oja, E. F.. Strong uniqueness of the extension of linear continuous functionals according to the Hahn Banach theorem. Mat. Zametki, 43 (1988), 237–246 (in Russian) = Math. Notes, 43 (1988), 134–139.Google Scholar
15.Oja, E.. Dual de 1'espace des opérateurs linéaires continus. C.R. Acad. Sci. Paris, 309, Ser. 1 (1989), 983–986.Google Scholar
16.Oja, E. and Pôldvere, M.. On subspaces of Banach spaces where every functional has a unique norm-preserving extension. Studia Math, 117 (1996), 289–306.Google Scholar
17.Phelps, R. R.. Uniqueness of Hahn-Banach extensions and unique best approximation. Trans Amer. Math. Soc., 95 (1960), 238–255.Google Scholar
18.Singer, I.. Bases in Banach spaces, Vol. 2 (Springer-Verlag, 1981).Google Scholar
19.Sullivan, F.. Geometric properties determined by the higher duals of a Banach space. Illinois J. Math., 21 (1977), 315–331.CrossRefGoogle Scholar
20.Taylor, A. E.. The extension oflinear functionals. Duke Math. J., 5 (1939), 538–547.CrossRefGoogle Scholar
21.Yost, D.. Approximation by compact operators between C(X) spaces. J. Approx. Th., 49 (1987), 99–109.Google Scholar