Disjointness preserving operators between little Lipschitz algebras

Given a real number α∈(0,1)α∈(0,1) and a metric space (X,d)(X,d), let Lipα(X)Lipα(X) be the algebra of all scalar-valued bounded functions f on X such thatpα(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‖∞}‖f‖=max{pα(f),‖f‖∞} or ‖f‖=pα(f)+‖f‖∞‖f‖=pα(f)+‖f‖∞. The little Lipschitz algebra lipα(X)lipα(X) is the closed subalgebra of Lipα(X)Lipα(X) formed by all those functions f   such that |f(x)−f(y)|/dα(x,y)→0|f(x)−f(y)|/d(x,y)α→0 as d(x,y)→0d(x,y)→0. A linear mapping T:lipα(X)→lipα(Y) is called disjointness preserving if f⋅g=0f⋅g=0 in lipα(X)lipα(X) implies (Tf)⋅(Tg)=0(Tf)⋅(Tg)=0 in lipα(Y)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)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)lipα(Y) and φ is a Lipschitz homeomorphism from Y onto X and, in particular, we obtain that T   is automatically continuous and T−1T−1 is disjointness preserving. Moreover we show that there exists always a discontinuous disjointness preserving linear functional on lipα(X)lipα(X), provided X is an infinite compact metric space.