Weakly peripherally multiplicative surjections of pointed Lipschitz algebras

Abstract

For (X,d) a compact metric space with a distinguished base point e_X let Lip_0(X) denote the Banach algebra of all scalar Lipschitz functions f on X such that f(e_X)=0, endowed with the norm L(f)={|f(x)−f(y)|/d(x,y): x,y∈X, x≠y}. Let ϕ:Lip_0(X)→Lip_0(Y) be a surjection such that Ran_π(fg)∩Ran_π(ϕ(f)ϕ(g))≠∅ for all f,g∈Lip_0(X), where Ran_π(f)={f(x): x∈X, |f(x)|=∥f∥_∞} is the peripheral range of f. Such a map ϕ is called weakly peripherally multiplicative. The main result of this paper shows that every weakly peripherally multiplicative map ϕ is a weighted composition operator of the form ϕ(f)(y)=τ(y)f(φ(y)) for all f∈Lip_0(X) and all y∈Y, where τ is a function from Y into {−1,1} and φ is a Lipschitz homeomorphism from Y onto X such that φ(e_Y)=e_X.