Automatic continuity of biseparating maps. (English) Zbl 1056.46032

Let \(X\), \(Y\) be realcompact spaces, \(E\) and \(F\) normed spaces, \(C(X,E)\) the set of all continuous \(E\)-valued functions on \(X\) and \(C_b(X,E)\) the space of all bounded continuous functions from \(X\) into \(E\). A linear map \(T: C(X,E) \to C(Y,F)\) is called separating if \(\| f(x)\| \;\| g(x)\| =0\) for every \(x\in E\) implies that \(\| (Tf)(y)\| \;\| (Tg)(y)\| =0\) for every \(y\in Y\), and biseparating if \(T^{-1}\) exists and is separating as well. One result of the paper is that every linear biseparating map \(T:C_b(X,E)\to C_b(Y,F)\) is continuous provided that \(Y\) has no isolated points. Another result tells us that if additionally \(E\) and \(F\) are infinite-dimensional, then a linear biseparating map \(T: C(X,E)\to C(Y,F)\) is continuous if the interior of the set of \(P\)-points of \(Y\) is empty. This is the best possible result.


46E40 Spaces of vector- and operator-valued functions
47B33 Linear composition operators
46H40 Automatic continuity
47B38 Linear operators on function spaces (general)
46E25 Rings and algebras of continuous, differentiable or analytic functions
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