Generation of Uniformly Closed Algebras of Functions

Generation of Uniformly Closed Algebras of Functions For a linear sublattice ℱ of C(X), the set of all real continuous functions on the completely regular space X, we denote by A(ℱ) the smallest uniformly closed and inverse-closed subalgebra of C(X) that contains ℱ. In this paper we study different methods to generate A(ℱ) from ℱ. For that, we introduce some families of functions which are defined in terms of suprema or sums of certain countably many functions in ℱ. And we prove that A(ℱ) is the uniform closure of each of these families. We obtain, in particular, a generalization of a known result about the generation of A(ℱ) when ℱ is a uniformly closed linear sublattice of bounded functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Generation of Uniformly Closed Algebras of Functions

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2005 by Springer
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-003-8543-y
Publisher site
See Article on Publisher Site

Abstract

For a linear sublattice ℱ of C(X), the set of all real continuous functions on the completely regular space X, we denote by A(ℱ) the smallest uniformly closed and inverse-closed subalgebra of C(X) that contains ℱ. In this paper we study different methods to generate A(ℱ) from ℱ. For that, we introduce some families of functions which are defined in terms of suprema or sums of certain countably many functions in ℱ. And we prove that A(ℱ) is the uniform closure of each of these families. We obtain, in particular, a generalization of a known result about the generation of A(ℱ) when ℱ is a uniformly closed linear sublattice of bounded functions.

Journal

PositivitySpringer Journals

Published: Jan 5, 2003

References

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