A generalized Banach-Stone theorem

Abstract

xeX (We recall that if X is compact, then C0(X) consists of all continuous complex-valued functions on X.) The well known Banach-Stone theorem states that if X and Y are locally compact Hausdorff spaces, and if C0(X) and C0(Y) are isometrically isomorphic, then X and Y are homeomorphic. The purpose of this article is to show that, with the additional hypothesis of first countability (which may well be unessential), the conclusion of the theorem remains true for a class of mappings somewhat more general than isometries. We wish to prove the following: THEOREM. If X and Y are first countable, locally compact Hausdorff spaces, and 4 is a continuous, norm-increasing linear isomorphism of Co(X) onto Co(Y) with bound strictly less than two, lIf!! _ lf+(f)ff < 1lfif lIf!, f E CO(X), 1ffij < 2,