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The strong closure of Boolean algebras of projections in Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear spaces and algebras of operators

Published online by Cambridge University Press:  09 April 2009

J. Diestel  and
W. J. Ricker
J. Diestel
Affiliation:
Department of Mathematical Sciences, Kent State University, P.O. Box 5190, Kent OH 44242-0001, USA e-mail: [email protected]
W. J. Ricker
Affiliation:
Math.-Geogr. Fakultät, Katholische Universität, Eichstätt-Ingolstadt, D-85072 Eichstätt, Germany e-mail: [email protected]
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Abstract

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This note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

MSC classification

Secondary: 46B25: Classical Banach spaces in the general theory 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 3 , December 2004 , pp. 365 - 370
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Dunford, N. and Schwartz, J. T., Linear operators III: spectral operators (Wiley-Interscience, New York, 1971).Google Scholar
[2]Gillespie, T. A., ‘Spectral measures on spaces not containing ℓ∞’, Proc. Edinburgh Math. Soc. (Ser. II) 24 (1981), 41–45.CrossRefGoogle Scholar
[3]Gillespie, T. A., ‘Strongly closed bounded Boolean algebras of projections’, Glasgow Math. J. 22 (1981), 73–75.CrossRefGoogle Scholar
[4]Gillman, L. and Jerison, M., Rings of continuous functions (van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
[5]Grothendieck, A., ‘Sur les applications lineaires faiblement compacts d'espaces du type C(K)’, Canad. J. Math. 5 (1953), 129–173.CrossRefGoogle Scholar
[6]Meyer-Nieberg, P., Banach lattices (Springer, Berlin, 1991).CrossRefGoogle Scholar
[7]Ricker, W. J., ‘The strong closure of σ-complete Boolen algebras of projections’, Archiv Math. (Basel) 72 (1999), 282–288.CrossRefGoogle Scholar
[8]Rosenthal, H. P., ‘On relatively disjoint families of measures with some applications to Banach space theory’, Studia Math. 37 (1970), 13–36.CrossRefGoogle Scholar