Non-affine functions and realcompact spaces

Abstract

Let and X be a topological space. An identity preserving a bounded map is called a ψ-homomorphism if is additive and . We call ψ a realcompact function if, whenever X is a realcompact space, any ψ-homomorphism is an evaluation at some point of X. By classical results of Hewitt and Shirota, respectively, the square as well as the absolute value are examples of realcompact functions. This paper extends these results and gives a complete description of realcompact functions. Indeed, it turns out that is a realcompact function if and only if ψ is non-affine. This leads to a Banach-Stone type theorem, namely, the realcompact spaces X and Y are homeomorphic if and only if and are ψ-isomorphic for some non-affine function .