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The uniform separation property and Banach-Stone theorems for lattice-valued Lipschitz functions. (English) Zbl 1202.46043
Proc. Am. Math. Soc. 137, No. 11, 3769-3777 (2009); correction ibid. 138, No. 4, 1535 (2010).
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 \(\max\{\text{Lip}(f),\|f\|_{\infty}\}\). 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 \( T: A(X,E) \rightarrow A(Y,F) \) be a vector lattice isomorphism that preserves the nowhere vanishing functions in both the directions. Then there exists a bi-Lipschitz map \(\phi: Y \rightarrow X\) and a Lipschtz map \(T^{\wedge}: Y \rightarrow L(E,F)\) such that \(T^{\wedge}\) takes values as lattice isomorphisms and \(T(f)(y)=T^{\wedge}(y)(f(\phi(y))\) for \(y \in Y\) and \(f \in A(X,E)\).

MSC:
46E40 Spaces of vector- and operator-valued functions
46E05 Lattices of continuous, differentiable or analytic functions
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