Disjointness preserving and local operators on algebras of differentiable functions

Abstract

This article is to discuss the automatic continuity properties and the representation of disjointness preserving linear mappings on certain normal Fréchet algebras of complex-valued functions. This class of operators is defined by the condition that any pair of functions with disjoint cozero sets is mapped to functions with disjoint cozero sets, and subsumes the class of local operators. It turns out that such operators are always continuous outside some finite singularity set of the underlying topological space. Our main emphasis is on disjointness preserving operators from Fréchet algebras of differentiable functions. Such operators are shown to admit a canonical representation that involves weighted composition for the derivatives. This result extends the classical characterization of local operators as linear partial differential operators.