Lipschitz algebras and peripherally-multiplicative maps

Abstract

Let X be a compact metric space and let Lip(X) be the Banach algebra of all scalar-valued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip(X), σ_π(f) denotes the peripheral spectrum of f. We state that any map Φ from Lip(X) onto Lip(Y ) which preserves multiplicatively the peripheral spectrum is a weighted composition operator of the form Φ(f) = τ · (f ◦ ϕ) for all f ∈ Lip(X), where τ : Y →{−1, 1} is a Lipschitz function and ϕ : Y → X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip(X)-algebras is of the form above.