Variations on the Banach-Stone theorem

Abstract

This paper is based on a series of lectures delivered during a 2001 summer course of the University of Cantabria in Spain. The central theme is the characterization of a topological space X in terms of the topological-algebraic structure of suitably chosen subspaces of the space C(X) of continuous functions on X . A huge variety of corresponding results is presented. After having discussed the classical Banach-Stone theorem, the authors present several more recent results which characterize a locally compact space X through the isometric/isomorphic structure of particular subspaces of C 0 (X) , the space of continuous functions on X which vanish at infinity. Another type of results allows to recover a complete metric space from spaces of bounded uniformly continuous functions, or of Lipschitz continuous functions, which take values in particular Banach spaces. Moreover, the paper contains a number of results regarding the characterization of a space X through algebraic properties of appropriate subspaces of C(X) , e.g., certain subalgebras. The problem of when isomorphy of spaces of differentiable functions on Banach manifolds entails isomorphy of the underlying manifolds is discussed in detail. The final chapter is devoted to the problem to decide when, for complete metric spaces X and Y , the existence of an isomorphism between suitable lattices of functions of uniformly continuous functions on X and Y , respectively, entails that X and Y are uniformly homeomorphic (similar for Lipschitz maps).