Lattices of uniformly continuous functions

An explicit representation of the order isomorphisms between lattices of uniformly continuous functions on complete metric spaces is given. It is shown that every lattice isomorphism T:U(Y)→U(X) is given by the formula (Tf)(x)=t(x,f(τ(x))), where τ:X→Y is a uniform homeomorphism and t:X×R→R is defin...

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Bibliographic Details
Published in:Topology and its applications 2013-01, Vol.160 (1), p.50-55
Main Authors: Cabello Sánchez, Félix, Cabello Sánchez, Javier
Format: Article
Language:English
Subjects:
Banach–Stone theorem
Isomorphism
Lattices
Uniformly continuous functions
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Summary:An explicit representation of the order isomorphisms between lattices of uniformly continuous functions on complete metric spaces is given. It is shown that every lattice isomorphism T:U(Y)→U(X) is given by the formula (Tf)(x)=t(x,f(τ(x))), where τ:X→Y is a uniform homeomorphism and t:X×R→R is defined by t(x,c)=(Tc)(x). This provides a correct proof for a statement made by Shirota sixty years ago.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2012.09.010