Automatic continuity of certain linear isomorphisms

Let S and T be compact, 0-dimensional, Hausdorff spaces and let C(S) and C(T) denote the sup-normed Banach spaces of continuous maps on S. and T, respectively, into a complete nonarchimedean valued field F. If F were the reals, then S and T would be homeomorphic if and only if C(S) and C(T) were isometrically isomorphic ; moreover, the isometric isomorphism A : C(T) C(S) must be of the following type : For each x in C(T) and any s in S, Ax(s) = a(s)x(h(s)) for some homeomorphism A of S onto T and some nonva-nishing continuous map a defined on S. As we have shown earlier, there are other kinds of isometric isomorphisms when dealing with F-valued functions. In this paper we focus attention on "separating" linear maps A, maps with the property that the cozero sets of Ax and Ay must be disjoint if the cozero sets of x and y are. We develop criteria for separating maps A to be continuous, for S and T to be homeomorphic, and for such maps to be of the "change of variables" type of the Stone-Banach theorem.Narici Lawrence, Beckenstein Edward. Automatic continuity of certain linear isomorphisms. In: Bulletin de la Classe des sciences, tome 73, 1987. pp. 191-200