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: ‖ T ( f ) T ( g ) ¯ − 1 ‖ ∞ = ‖ f g ¯ − 1 ‖ ∞ ( f , g ∈ Lip 0 ( X ) ) , then T is of the form T ( f ) = τ ⋅ ( η ⋅ ( f ○ φ ) + ( 1 − η ) ⋅ ( f ○ φ ) ¯ ) ( f ∈ Lip 0 ( X ) ) , where η and τ are functions on Y such that 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: Ran π ( f g ¯ ) ∩ Ran π ( T ( f ) T ( g ) ¯ ) ≠ ∅ ( f , g ∈ Lip 0 ( X ) ) , where Ran π ( f ) denotes the peripheral range of f, then T can be expressed as T ( f ) = τ ⋅ ( f ○ φ ) ( f ∈ Lip 0 ( X ) ) , with τ and φ as above. As a consequence, we obtain similar descriptions for surjective maps between Lipschitz ∗-algebras Lip ( X ) .