Non‐affine functions and realcompact spaces

Let ψ ∈ C ( R ) and X be a topological space. An identity preserving a bounded map h : C ( X ) → R is called a ψ‐ homomorphism if h is additive and h ∘ ψ = ψ ∘ h . We call ψ a realcompact function if, whenever X is a realcompact space, any ψ‐homomorphism h : C ( X ) → R 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 ψ ∈ C ( R ) 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 C ( X ) and C ( Y ) are ψ‐isomorphic for some non‐affine function ψ ∈ C ( R ) .