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A representation theorem for a complete Boolean algebra of projections*

Published online by Cambridge University Press:  14 November 2011

H. R. Dowson  and
T. A. Gillespie
H. R. Dowson
Affiliation:
Department of Mathematics, University of Glasgow
T. A. Gillespie
Affiliation:
Department of Mathematics, University of Edinburgh
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Synopsis

Let B be a complete Boolean algebra of projections on a complex Banach space X and let (B) denote the closed algebra of operators generated by B in the norm topology. It is shown that there is a complex Hilbert space H, a complete Boolean algebra B0 of self-adjoint projections on H, and an algebraic isomorphism of B onto B. This isomorphism is bicontinuous when B and B are endowed with the norm topologies, the weak operator topologies or the ultraweak operator topologies. It is also bicontinuous on bounded sets with respect to the strong operator topologies on B and B. As an application, it is shown that the weak and ultraweak operator topologies in fact coincide on B.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 3-4 , 1979 , pp. 225 - 237
Copyright
Copyright © Royal Society of Edinburgh 1979

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