Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T : Lip(0)(X) -> Lip(0)(Y) be a surjective map between pointed Lipschitz *-algebras. where X and Y are compact metric spaces. On the one hand, we prove that if T satisfies the non-symmetric norm *-multiplicativity condition: parallel to T(f)<(T(g))over bar> - 1 parallel to(infinity) = parallel to f (g) over bar - 1 parallel to(infinity) (f, g is an element of Lip(0)(X)), then T is of the form T(f) = tau.(eta.(f circle phi) + (1 - eta).<((f circle phi))over bar>) (f is an element of Lip(0)(X)), where eta and tau are functions on Y such that eta(Y) subset of {0, 1} and tau(Y) subset of {alpha is an element of K: vertical bar alpha vertical bar = 1}, and phi : Y -> X is a base point preserving Lipschitz homeomorphism. On the other hand, if T satisfies the weakly peripherally *-multiplicativity condition: Ran(pi)(f (g) over bar) boolean AND Ran(pi) (T(f)<(T(g))over bar>) not equal 0 (f, g is an element of Lip(0)(X)), where Ran(pi)(f) denotes the peripheral range of f, then T can be expressed as T(f) = tau . (f circle phi) (f is an element of Lip(0)(X)), with tau and phi as above. As a consequence, we obtain similar descriptions for surjective maps between Lipschitz *-algebras Lip(X). (C) 2009 Elsevier Inc. All rights reserved.