A geometric characterization of ring homomorphisms on \(f\)-rings (Q2847686)

Let \(A\) be an \(f\)-ring with identity and \(B\) an Archimedean \(f\)-ring. The authors consider group homomorphisms from \(A\) to \(B\) that map the identity to a fixed idempotent element \(w\) in \(B\). They show that such a mapping is a ring homomorphism if and only if it is an extreme point of the set of all such maps. They derive a characterization of such ring homomorphisms in terms of a Gelfand-type transform. They finally show that in case \(u=1\), the homomorphisms are (up to multiplicative constants) the basis elements of the \(\ell\)-group of all bounded group homomorphisms from \(A\) to \(\mathbb R\).