Homomorphisms with respect to a function (Q431174)

Let \(X\) be a realcompact space, \(C(X)\) the space of continuous real-valued functions on \(X\), and \(H:C(X)\to \mathbb R\) be an identity and order preserving homomorphism. The authors show that \(H\) is an evaluation mapping (i.e. there is some \(x\in X\) so that for all \(f\in C(X)\), \(H(f)=f(x)\)) if and only if there exists \(\varphi\in C(\mathbb R)\) with \(\varphi(r)>\varphi(0)\) for all \(r\neq 0\) for which \(H\circ \varphi=\varphi\circ H\). This extends and unifies classical results of Hewitt and Shirota, respectively.