Inverses of disjointness preserving operators

Abstract

A linear operator between (possibly vector-valued) function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. Here, two functions are said to be disjoint if at each point at least one of them vanishes. In this paper, we study linear disjointness preserving operators between various types of function spaces, including spaces of (little) Lipschitz functions, uniformly continuous functions and differentiable functions. It is shown that a disjointness preserving linear isomorphism whose domain is one of these types of spaces (scalar-valued) has a disjointness preserving inverse, subject to some topological conditions on the range space. A representation for a general linear disjointness preserving operator on a space of vector-valued C p functions is also given.