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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 Lip(X,E)={all bounded E-valued Lipschitz functions}; Lip(X)={all bounded Lipschitz functionals}; L ' (E,F)={all linear bijections from E to F}. A map T:Lip(X,E)Lip(Y,F) is said to be separating if T is linear and Tf(y)Tg(y)=0 for all yY, whenever f,gLip(X,E) satisfy fx<g(x)=0 for all xX. 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:Lip(X,E)Lip(Y,F) be a biseparating map. Then there exists a bi-Lipschitz homeomorphism h:YX and a map J:YL ' (E,F) such that Tf(y)=(Jy)(f(h(y))) for all fLip(X,E) and yY.

Proposition 2. Let T:Lip(X)Lip(Y) be a bijective separating map. If Y is compact, then T is biseparating and continuous.

47B38Operators on function spaces (general)
46E10Topological linear spaces of continuous, differentiable or analytic functions
54C35Function spaces (general topology)