Automatic continuity of biseparating maps. (English) Zbl 1056.46032

Araujo, Jesús; Jarosz, Krzysztof

Stud. Math. 155, No. 3, 231-239 (2003).

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.

Reviewer: Raymond Mortini (Metz)

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

Keywords:

automatic continuity of biseparating maps; realcompact spaces; P-points