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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 $\Vert f(x)\Vert \ \Vert g(x)\Vert =0$ for every $x\in E$ implies that $\Vert (Tf)(y)\Vert \ \Vert (Tg)(y)\Vert =0$ 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.

46E40Spaces of vector- and operator-valued functions
47B33Composition operators
46H40Automatic continuity
47B38Operators on function spaces (general)
46E25Rings and algebras of continuous, differentiable or analytic functions
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