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Biseparating maps between Lipschitz function spaces. (English) Zbl 1169.47024
Let \(X,Y\) be bounded complete metric spaces and let \(E,F\) be (real or complex) normed spaces. We write \(\text{Lip}(X,E)= \{\)all bounded \(E\)-valued Lipschitz functions}; \(\text{Lip}(X)= \{\)all bounded Lipschitz functionals}; \(L'(E,F)=\{\)all linear bijections from \(E\) to \(F\}\). A map \(T:\text{Lip}(X,E)\to \text{Lip}(Y,F)\) is said to be separating if \(T\) is linear and \(\|Tf(y)\|\,\|Tg (y)\|=0\) for all \(y\in Y\), whenever \(f,g\in \text{Lip}(X,E)\) satisfy \(\|fx<\|\|g(x) \|=0\) for all \(x\in X\). \(T\) is said to be biseparating if \(T\) is bijective and both \(T\) and \(T^{-1}\) are separating. The authors establish the following results.
Proposition 1. Let \(T:\text{Lip}(X,E)\to \text{Lip}(Y,F)\) be a biseparating map. Then there exists a bi-Lipschitz homeomorphism \(h:Y \to X\) and a map \(J:Y\to L'(E,F)\) such that \(Tf(y)=(Jy) (f(h(y)))\) for all \(f\in \text{Lip}(X,E)\) and \(y\in Y\).
Proposition 2. Let \(T:\text{Lip}(X)\to \text{Lip}(Y)\) be a bijective separating map. If \(Y\) is compact, then \(T\) is biseparating and continuous.

MSC:
47B38 Linear operators on function spaces (general)
46E10 Topological linear spaces of continuous, differentiable or analytic functions
54C35 Function spaces in general topology
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