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 ≠ 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 ( T f ) ⋅ ( T g ) = 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 T f = 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.