The uniform separation property and Banach–Stone theorems for lattice-valued Lipschitz functions

Abstract

Abstract. Using the uniform separation property of N. Weaver and the uniform joint property, we present in this paper a Lipschitz version of a BanachStone-type theorem for lattice-valued continuous functions obtained recently

Common zeros preserving maps on vector-valued function spaces and Banach modules

“…For some results concerning non-vanishing preserving maps (on lattices) one can see [4,6,5,8,9,15,18,19,20]. We also refer to [14] for some relevant concepts in the case of scalar-valued continuous functions.…”

Nonvanishing Preservers and Compact Weighted Composition Operators between Spaces of Lipschitz Functions

“…Later, Jiménez-Vargas and Villegas-Vallecillos [12] proved that two little Lipschitz algebras are order isomorphic if and only if the corresponding compact metric spaces are Lipschitz homeomorphic. Recently, Jiménez-Vargas et al [13] presented a Lipschitz version of the result in [5], in which the underlying spaces should be compact.…”

Maps preserving common zeros between subspaces of vector-valued continuous functions

“…Finally, in [4], it was proved that X and Y are homeomorphic and E and F are Riesz isomorphic whenever T : C(X, E) → C(Y, F ) is a Riesz isomorphism preserving non-vaninshing functions, X and Y are compact Hausdorff spaces, and E and F are arbitrary Banach lattices. Recently, the Lipschitz version of this topic has been obtained in [9], providing a complete characterization of vector lattice isomorphisms T : A(X, E) → A(Y, F ) preserving non-vanishing functions, where X and Y are compact metric spaces, E and F are Banach lattices, and A(X, E) and A(Y, F ) are vector sublattices of Lip(X, E) and Lip(Y, F ), respectively, that separate and join points uniformly.…”