Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Abstract

Let (X,d_X) and (Y,d_Y) be pointed compact metric spaces with distinguished base points e_X and e_Y. The Banach algebra of all scalar-valued Lipschitz functions on X that map the basepoint e_X to 0 is denoted by Lip_0(X). The peripheral range of a function f in Lip_0(X) is the set Ran(f)={f(x) : |f(x)|=||f||_\infty} of range values of maximum modulus. We prove that if T_1,T_2 : Lip_0(X)->Lip_0(Y) and S_1,S_2 : Lip_0(X)->Lip_0(X) are surjective mappings such that Ran_\pi(T_1(f)T_2(g))\cap Ran_\pi(S_1(f)S_2(g))\neq \emptyset for all f,g in Lip_0(X), then there are mappings \varphi_1, \varphi_2 : Y->K with \varphi_1(y)\varphi_2(y)=1 for all y in Y and a basepoint-preserving Lipschitz homeomorphism \psi : Y->X such that T_j(f)(y)=\varphi_j(y)S_j(f)(\psi(y)) for all f in Lip_0(X), y in Y , and j = 1, 2. In particular, if S_1 and S_2 are identity functions, then T_1 and T_2 are weighted composition operators.