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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:
 4.6e+41 Spaces of vector- and operator-valued functions 4.6e+06 Lattices of continuous, differentiable or analytic functions
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