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Strongly closed bounded Boolean algebras of projections

Published online by Cambridge University Press:  18 May 2009

T. A. Gillespie
T. A. Gillespie
Affiliation:
Department of Mathematics, James Clerk Maxwell Building, The King's Buildings, Edinburgh EH9 3JZ
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It is known that every complete Boolean algebra of projections on a Banach space X is strongly closed and bounded and that, although the converse of this result fails in general, it is valid if X is weakly sequentially complete [1, XVII. 3, pp. 2194–2201]. In the present note it is shown that this converse is in fact valid precisely when X contains no subspace isomorphics to the sequence space c0. More explicitly, the following two results are proved. In both, X may be a real or complex space, but c0 will consist of the null sequences in the underlying scalar field.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 1 , January 1981 , pp. 73 - 75
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

REFERENCES

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3.Tzafriri, L., Reflexivity in Banach lattices and their subspaces, J. Functional Analysis 10 (1972), 1–18.Google Scholar