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Common zeros preserving maps on vector-valued function spaces and Banach modules. (English) Zbl 1358.46048

Summary: Let \(X\), \(Y\) be Hausdorff topological spaces, and let \(E\) and \(F\) be Hausdorff topological vector spaces. For certain subspaces \(A(X, E)\) and \(A(Y,F)\) of \(C(X,E)\) and \(C(Y,F)\) respectively (including the spaces of Lipschitz functions), we characterize surjections \(S,T: A(X,E) \to A(Y,F)\), not assumed to be linear, which jointly preserve common zeros in the sense that \(Z(f-f^\prime) \cap Z(g-g^\prime) \neq \varnothing\) if and only if \(Z(Sf-Sf^\prime) \cap Z(Tg-Tg^\prime) \neq \varnothing\) for all \(f,f^\prime,g,g^\prime \in A(X,E)\). Here \(Z(\cdot)\) denotes the zero set of a function. Using the notion of point multipliers we extend the notion of zero set for the elements of a Banach module and give a representation for surjective linear maps which jointly preserve common zeros in module case.

MSC:

46J10 Banach algebras of continuous functions, function algebras
47B48 Linear operators on Banach algebras
46J20 Ideals, maximal ideals, boundaries
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