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Wallman Compactification and Representation

Published online by Cambridge University Press:  20 November 2018

Shankar Hegde
Shankar Hegde*
Affiliation:
Northern Illinois University, DeKalb, Illinois
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Let X be any set and A be a uniformly closed algebra of bounded real valued functions on X which contains the constants and separates the points. For a lattice ℒ of subsets of X (we assume throughout that ∅ and X belong to ℒ), let MR(ℒ) denote the space of all finite, finitely additive,ℒ-regular measures defined on the field of sets generated by ℒ . Generalizing the notion of an integral representation, in [5] Kirk and Crenshaw define a standard representation of A*, the Banach dual of A, in MR(ℒ) to be a linear map I of A* into MR(ℒ) with the property that if 0 ≦ ϕ ∈ A*, then

for every W in ℒ. The space MR(ℒ) is said to represent A* if there exists a (unique) standard representation I of A* onto MR(ℒ) which is a Banach lattice isomorphism.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 372 - 380
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Brooks, R. M., On Wallman compactifications, Fund. Math. GO (1967), 157-173.Google Scholar
2. Dunford, N. and Schwartz, J. T., Linear operators, Part I (Interscience, New York, 1958).Google Scholar
3. Frink, O., Compactifications and seminormal spaces, Amer. J. Math. 86 (1964), 602–607.Google Scholar
4. Hegde, S., A characterization of z-separating algebras, Proc. Amer. Math. Soc. 73 (1979), 40–44.Google Scholar
5. Kirk, R. B. and Crenshaw, J. A., A generalized topological measure theory, Trans. Amer. Math. Soc. 207 (1975), 189–217.Google Scholar
6. Sultan, A., Measure, compactification and representation, Can. J. Math. 30 (1978), 54–65.Google Scholar
7. Varadarajan, V. S., Measures on topological spaces, Amer. Math. Soc. Transi. 48 (1965), 161–228.Google Scholar