A Banach-Stone Theorem for Uniformly Continuous Functions

Homomorphisms on Lattices of Continuous Functions

“…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.…”

“…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.…”

“…As in [12] we consider only metric spaces. We abandon our earlier convention that X and Y denote compact spaces and, given a metric space X, we write U * (X) for the lattice of all bounded uniformly continuous funtions on X.…”

Lipschitz-type functions on metric spaces

“…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).…”

“…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]).…”