Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T:Lip0(X)→Lip0(Y)T:Lip0(X)→Lip0(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:‖T(f)T(g)¯−1‖∞=‖fg¯−1‖∞(f,g∈Lip0(X)), then T is of the formT(f)=τ⋅(η⋅(f○φ)+(1−η)⋅(f○φ)¯)(f∈Lip0(X)), where η and τ are functions on Y   such that η(Y)⊆{0,1}η(Y)⊆{0,1} and τ(Y)⊆{α∈K:|α|=1}, and φ:Y→X is a base point preserving Lipschitz homeomorphism. On the other hand, if T satisfies the weakly peripherally ∗-multiplicativity condition:Ranπ(fg¯)∩Ranπ(T(f)T(g)¯)≠∅(f,g∈Lip0(X)), where Ranπ(f)Ranπ(f) denotes the peripheral range of f, then T can be expressed asT(f)=τ⋅(f○φ)(f∈Lip0(X)), with τ and φ as above. As a consequence, we obtain similar descriptions for surjective maps between Lipschitz ∗-algebras Lip(X)Lip(X).