2000
DOI: 10.1007/s006050070008
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A Banach-Stone Theorem for Uniformly Continuous Functions

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Cited by 26 publications
(22 citation statements)
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“…As it is remarked in [12], each metric space (X, d) admits a compactification κX such that the Banach algebras U * (X) and C(κX) are isometrically isomorphic in the obvious way. What is more important [12,Lemma 1]: if X is complete, then the points of κX having countable neighborhood bases are exactly those of X.…”
Section: Uniformly Continuous Functionsmentioning
confidence: 97%
See 2 more Smart Citations
“…As it is remarked in [12], each metric space (X, d) admits a compactification κX such that the Banach algebras U * (X) and C(κX) are isometrically isomorphic in the obvious way. What is more important [12,Lemma 1]: if X is complete, then the points of κX having countable neighborhood bases are exactly those of X.…”
Section: Uniformly Continuous Functionsmentioning
confidence: 97%
“…What is more important [12,Lemma 1]: if X is complete, then the points of κX having countable neighborhood bases are exactly those of X.…”
Section: Uniformly Continuous Functionsmentioning
confidence: 99%
See 1 more Smart Citation
“…Note that, in the class of complete metric spaces X, the following function spaces determine the corresponding structure on X, namely, C(X) and C * (X) the topological structure (see Gillman and Jerison [11]), U(X) and U * (X) the uniform structure (see [6]), and, as we have seen in above Theorem 2, LS(X) and LS * (X) the LS-structure. In addition, according to [8], we can say that Lip(X) determines the Lispchitz structure on X, but the same is not true for Lip * (X) (a simple example is obtained by considering (X, d) and (X, inf{1, d}), where d is unbounded).…”
Section: Small-determined Metric Spacesmentioning
confidence: 99%
“…Finally, in order to see that (c) implies (a), we need to make use of the structure space H (L) associated to a unital vector lattice L, which is defined as the space of all real-valued unital vector lattice homomorphisms on L endowed with the pointwise topology, that is, the topology that inherits as a subspace of the product R L (see [6] or [8]). Thus, we have that H (LS * (X)) = H (Lip * (X)) is a compact topological space containing X densely, and we know that a point in H (LS * (X)) has a countable neighborhood basis if, and only if, it belongs to X (see [8]).…”
Section: Theorem 2 (Banach-stone Type)mentioning
confidence: 99%