Order isomorphisms of little Lipschitz algebras

Abstract

For compact metric spaces (X,d_X) and (Y,d_Y) and scalars \alpha,\beta in (0,1), we prove that every order isomorphism T between little Lipschitz algebras lip(X,d_X^\alpha) and lip(Y,d_Y^\beta) is a weighted composition operator of the form T(f)(y)=a(f(h(y)) for all f in lip(X,d_X^\alpha) and y in Y, where a is a nonvanishing positive function in lip(Y,d_Y^\beta) and h is a Lipschitz homeomorphism from (Y,d_Y^\beta) onto (X,d_X^\alpha).