Memorandum on multiplicative bijections and order

Abstract

Let C(X,I) denote the semigroup of continuous functions from the topo- logical space X to I =( 0, 1), equipped with the pointwise multiplication. The pa- per studies semigroup homomorphisms C(Y,I) → C(X,I), with emphasis on iso- morphisms. The crucial observation is that, in this setting, homomorphisms pre- serve order, so isomorphisms preserve order in both directions and they are auto- matically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete met- ric spaces of finite diameter without isolated points, every multiplicative bijection T : Lip(Y, I) → Lip(X, I) has the form Tf = f ◦ τ , where τ : X → Y is a Lipschitz homeomorphism.