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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

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