Lipschitz Algebras and Peripherally-multiplicative Maps

<正> 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）,σπ（f）denotesthe peripheral spectrum of f.We state that any map Φ from Lip（X）onto Lip（Y）which preservesmultiplicatively the peripheral spectrum:σπ(Φ（f）Φ（g）)=σπ（fg）,∨f,g∈Lip（X）is a weighted composition operator of the form Φ（f）=Υ·（foΦ）for all f ∈ Lip（X）,where Υ:Y→{-1,1}is a Lipschitz function andΦ:Y→X is a Lipschitz homeomorphism.As a consequence ofthis result,any multiplicatively spectrum-preserving surjective map between Lip（X）-algebras is of theform above.