Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Antonio Jiménez-Vargas 1 , Kristopher Lee 2 , Aaron Luttman 3 ,  and Moisés Villegas-Vallecillos 4
  • 1 Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Almería, 04120, Spain
  • 2 Department of Mathematics, Iowa State University, Ames, IA, 50011, USA
  • 3 Mathematics and Software Development, National Security Technologies, LLC, P.O. Box 98521, M/S NLV078, Las Vegas, NV, 89193, USA
  • 4 Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, 11510, Spain

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 $\mathbb{K}$-valued Lipschitz functions on X — where $\mathbb{K}$ is either‒or ℝ — that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = {f(x): |f(x)| = ‖f‖∞} of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb{K}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ 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.

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