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Spectral measures on spaces not containing l∞

Published online by Cambridge University Press:  20 January 2009

T. A. Gillespie
T. A. Gillespie
Affiliation:
Department of MathematicsJames Clerk Maxwell BuildingThe King's BuildingsEdinburgh EH9 3JZ
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The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then

(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that

(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that

(iii) every prespectral operator on X is spectral.

(See also (1, Theorem 6.11, p. 165) for (iii).)

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 24 , Issue 1 , February 1981 , pp. 41 - 45
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

REFERENCES

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