Disjointness preserving operators between little Lipschitz algebras ✩

Given a real number α ∈ (0, 1) and a metric space (X, d), let Lipα(X) be the algebra of all scalar-valued bounded functions f on X such that pα(f ) = sup f (x)− f (y) /d(x, y)α: x,y ∈ X, x = y <∞, endowed with any one of the norms f =max{pα(f ), f ∞} or f =pα(f )+ f ∞. The little Lipschitz algebra lipα(X) is the closed subalgebra of Lipα(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 : lipα(X)→lipα(Y ) is called disjointness preserving if f · g = 0 in lipα(X) implies (Tf ) · (T g) = 0 in lipα(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 lipα(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 lipα(X), provided X is an infinite compact metric space. © 2007 Elsevier Inc. All rights reserved.