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On a theorem of Sobczyk

Published online by Cambridge University Press:  17 April 2009

Aníbal Moltó
Aníbal Moltó
Affiliation:
Departamento de Análisis Matematico, Universitat de Valencia, Dr. Moliner 50 46100 Burjassot, Valencia, Spain
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Abstract

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In this paper the result of Sobczyk about complemented copies of c0 is extended to a class of Banach spaces X such that the unit ball of their dual endowed with the weak* topology has a certain topological property satisfied by every Corson-compact space. By means of a simple example it is shown that if Corson-compact is replaced by Rosenthal-compact, this extension does not hold. This example gives an easy proof of a result of Phillips and an easy solution to a question of Sobczyk about the existence of a Banach space E, c0 ⊂ E ⊂ l∞, such that E is not complemented in l∞ and c0 is not complemented in E. Assuming the continuum hypothesis, it is proved that there exists a Rosenthal-compact space K such that C(K) has no projectional resolution of the identity.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 123 - 130
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Argyros, S., Mercourakis, S. and Negrepontis, S., ‘Functional-analytic properties of Corson-compact spaces’, Studia. Math. 89 (1988), 197–229.CrossRefGoogle Scholar
[2]Bourgain, J., Fremlin, D.H. and Talagrand, M., ‘Pointwise compact sets of Baire-measurable functions’, Amer. J. Math. 100 (1978), 845–886.CrossRefGoogle Scholar
[3]Deville, R. and Godefroy, G., ‘Some applications of projective resolutions of identity’, (preprint).Google Scholar
[4]Engelking, R., General Topology (PWN-Polish Scientific Publishers, Warsaw, 1977).Google Scholar
[5]Godefroy, G., ‘Compacts de Rosenthal’, Pacific J. Math. 91 (1980), 293–306.CrossRefGoogle Scholar
[6]Isbell, J. and Semadeni, Z., ‘Projection constants and spaces of continuous functions’, Trans. Amer. Math. Soc. 107 (1963), 38–48.CrossRefGoogle Scholar
[7]Jameson, G.J.O., Topology and normed spaces (Chapman and Hall, London, 1974).Google Scholar
[8]Pelczynski, A. and Sudakov, V.N., ‘Remark on non-complemented subspaces of the space m(S)’, Colloq. Math. 9 (1962), 85–88.CrossRefGoogle Scholar
[9]Phillips, R.S., ‘On Linear transformations’, Trans. Amer. Math. Soc. 48 (1940), 516–541.CrossRefGoogle Scholar
[10]Sobczyk, A., ‘Projection of the space (m) on its subspace (c 0)’, Bull. Amer. Math. Soc. 47 (1941), 938–947.CrossRefGoogle Scholar
[11]Valdivia, M., ‘Resolutions of the identity in certain Banach spaces’, Collect. Math. 39 (1988), 127–140.Google Scholar
[12]Veech, W.A., ‘Short proof of Sobczyk's theorem’, Proc. Amer. Math. Soc. 28 (1971), 627–628.Google Scholar
[13]Whitley, R., ‘Projecting m onto c 0’, Amer. Math. Monthly 73 (1966), 285–286.CrossRefGoogle Scholar