Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-27T14:26:28.633Z Has data issue: false hasContentIssue false
  • English
  • Français

Boolean algebras of projections

Published online by Cambridge University Press:  20 January 2009

P. G. Spain
P. G. Spain
Affiliation:
University of Glasgow, Glasgow G12 8QW
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 3 , March 1975 , pp. 287 - 289
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Bade, W. G., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345–360.CrossRefGoogle Scholar
(2) Barry, J. Y., On the convergence of ordered sets of projections, Proc. Amer. Math. Soc. 5 (1954), 313–314.CrossRefGoogle Scholar
(3) Bartle, R. G., Dunford, N. and Schwartz, J. T., Weak compactness and vector measures, Canadian J. Math. 7 (1955), 289–305.CrossRefGoogle Scholar
(4) Berkson, E., A characterization of scalar type operators on reflexive Banach spaces, Pacific J. Math. 13 (1963), 365–373.CrossRefGoogle Scholar
(5) Berkson, E., Some characterizations of C*-algebras, Illinois J. Math. 10 (1966), 1–8.CrossRefGoogle Scholar
(6) Bonsall, F. F. and Duncan, J., Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras (Cambridge University Press, 1971).CrossRefGoogle Scholar
(7) Dunford, N. and Schwartz, J. T., Linear Operators (Interscience, New York, 1958).Google Scholar
(8) Edwards, D. A. and Tulcea, C. T. Ionescu, Some remarks on commutative algebras of operators on Banach spaces, Trans. Amer. Math. Soc. 93 (1959), 541–551.CrossRefGoogle Scholar
(9) Spain, P. G., On commutative V*-algebras II, Glasgow Math. J. 13 (1972), 129–134.CrossRefGoogle Scholar