Let be a compact metric space, let be the set of complex or real numbers. The Banach space of all Lipschitz functions on to with the norm is denoted by . Here is the supremum norm and is the Lipschitz constant of . is a commutative Banach algebra with respect to pointwise multiplication, but it also is an ordered vector space with respect to the pointwise order defined by if and only if and for all . The little Lipschitz algebra is the closed subspace of consisting of all those functions in with the property that for each , there exists such that implies .
Let be a real number in , then by the authors denote the metric . The metric space and the Lipschitz algebras , are considered in the paper.
Let and be compact metric spaces, and let and be real numbers in . A linear map is called an order isomorphism if is bijective and both and are order-preserving. If is a function in and is a Lipschitz homeomorphism from onto , then the map defined by for every is an order isomorphism. The main result of the paper is the proof that the converse is also true: every order isomorphism from onto is a weighted composition operator of the form .