Lipschitz algebras and peripherally-multiplicative maps

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 is an element of Lip(X), sigma(pi)(f) denotes the peripheral spectrum of f. We state that any map Phi from Lip(X) onto Lip(Y) which preserves multiplicatively the peripheral spectrum: sigma(pi) (Phi(f)Phi(g)) = sigma(pi)(fg), for all f, g is an element of Lip (X), is a weighted composition operator of the form Phi(f) = tau center dot (f circle phi) for all f is an element of Lip(X), where tau : Y -> {-1, 1} is a Lipschitz function and phi : 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.