*(English)*Zbl 1202.46043

This paper contributes to a very active area of ‘local’ Banach-Stone theorems for spaces of vector-valued functions. For an isomorphism $T$ of such spaces, ‘local conditions’ are assumed on $T$ to ensure that the underlying spaces are isomorphic and $T$ is described in terms of these objects in a canonical way (composition operator).

Let $(X,d)$ be a metric space and let $E$ be a nonzero Banach lattice. Let Lip$(X,E)$ denote the Banach space of bounded $E$-valued Lipschitz functions with pointwise order and with respect to the norm $max\{\text{Lip}\left(f\right),\parallel f{\parallel}_{\infty}\}$. For $X,Y$ and $E,F$ in this category, let $A(X,E)$ and $A(Y,F)$ be *closed* sublattices (this correction gets noted in “Correction to ‘The uniform separation property and Banach-Stone theorems for lattice-valued Lipschitz functions”’, ibid. 138, 1535) such that they separate and join points uniformly. Let $T:A(X,E)\to A(Y,F)$ be a vector lattice isomorphism that preserves the nowhere vanishing functions in both the directions. Then there exists a bi-Lipschitz map $\phi :Y\to X$ and a Lipschtz map ${T}^{\wedge}:Y\to L(E,F)$ such that ${T}^{\wedge}$ takes values as lattice isomorphisms and $T\left(f\right)\left(y\right)={T}^{\wedge}\left(y\right)(f\left(\phi \left(y\right)\right)$ for $y\in Y$ and $f\in A(X,E)$.