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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.

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