Multiplicative semigroups of Lipschitz functions

Abstract

Given a (complete) metric space X, we denote by Lip(X) the space of real-valued Lipschitz functions on X and we equip it with the pointwise product. The purpose of this note is to describe those bijections T : Lip(Y ) → Lip(X) which are \multiplicative " in the sense that whenever f; g ∈ Lip(Y ) are such that fg ∈ Lip(Y ) one has T(fg) = T(f)T(g). The main result of the paper states that if X has no isolated points, then every mul- tiplicative bijection T : Lip(Y ) → Lip(X) arises as T(f) = f τ _ , where τ : X → Y is a Lipschitz homeomorphism and so it is automatically linear. We also give a description of the semigroup isomorphisms T : Lip(Y ) → Lip(X) in the case where the underlying metric spaces are compact.