## 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.

### MSC:

 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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