Lipschitz algebras and peripherally-multiplicative maps

Abstract

Let X be a compact metric space and let Lip(X) be the Banach algebra of all scalarvalued 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: $$ \sigma _\pi (\Phi (f)\Phi (g)) = \sigma _\pi (fg),\forall f,g \in Lip(X), $$ 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.