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Cyclic Banach spaces and reflexive operator algebras*

Published online by Cambridge University Press:  14 November 2011

T. A. Gillespie
T. A. Gillespie
Affiliation:
Department of Mathematics, University of Edinburgh
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Synopsis

Let X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ, the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 78 , Issue 3-4 , 1978 , pp. 225 - 235
Copyright
Copyright © Royal Society of Edinburgh 1978

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Footnotes

*

This paper was assisted in publication by a grant from the Carnegie Trust for the Universities of Scotland.

References

1 Arveson, W. B.. A density theorem for operator algebras. Duke Math. J. 34 (1967), 635–647.CrossRefGoogle Scholar
2 Berkson, E. and Dowson, H. R.. On reflexive scalar-type spectral operators. J. London Math. Soc. 8 (1974), 652–656.Google Scholar
3 Dunford, N. and Schwartz, J. T.. Linear operators III (New York: Wiley, 1971).Google Scholar
4 Halmos, P. R.. A Hilbert space problem book (Princeton: Van Nostrand, 1967).Google Scholar
5 Hoffman, K.. Banach spaces of analytic functions (Englewood Cliffs: Prentice-Hall, 1962).Google Scholar
6 Radjavi, H. and Rosenthal, P.. Invariant subspaces (Berlin: Springer, 1973).Google Scholar
7 Schaefer, H. H.. Banach lattices andpositive operators (Berlin: Springer, 1974).CrossRefGoogle Scholar
8 Veksler, A. I.. Cyclic Banach spaces and Banach lattices. Dokl. Akad. Nauk SSSR 213 (1973), 770–773. (Transl. Soviet Math. Dokl. 14 (1973), 1774–1779.Google Scholar
9 Zaanen, A. C.. Integration (Amsterdam: North-Holland, 1967).Google Scholar