Disjointness preserving operators between little Lipschitz algebras

Abstract

Given a real number α∈(0,1) and a metric space (X,d), let Lip_α(X) be the algebra of all scalar-valued bounded functions f on X such that p_α(f)=sup{|f(x)-f(y)|/d(x,y)^α: x,y ∈ X, x\neq y}<∞, endowed with any one of the norms ||f||=max{p_α(f),||f||_∞} or ||f||=p_α(f)+||f||_∞. The little Lipschitz algebra lip_α(X) is the closed subalgebra of Lip_α(X) formed by all those functions f such that |f(x)−f (y)|/d(x,y)^α→0 as d(x, y)→0. A linear mapping T:lip_α(X)→lip_α(Y) is called disjointness preserving if f · g = 0 in lip_α(X) implies (Tf) · (Tg)=0 in lip_α(Y). In this paper we study the representation and the automatic continuity of such maps T in the case in which X and Y are compact. We prove that T is essentially a weighted composition operator Tf = h · (f ◦ ϕ) for some nonvanishing little Lipschitz function h and some continuous map ϕ. If, in addition, T is bijective, we deduce that h is a nonvanishing function in lip_α(Y ) and ϕ is a Lipschitz homeomorphism from Y onto X and, in particular, we obtain that T is automatically continuous and T^{-1} is disjointness preserving. Moreover we show that there exists always a discontinuous disjointness preserving linear functional on lip_α(X), provided X is an infinite compact metric space.