Order Isomorphisms between Positive Cones of $C_0(X)$

Abstract

Let $X$ and $Y$ be locally compact Hausdorff spaces. We study order isomorphisms \[ T:C_0^+(X)\to C_0^+(Y), \] where $C_0(X)$ denotes the Banach space of all real-valued continuous functions on $X$ vanishing at infinity, and \[ C_0^+(X)=\{f\in C_0(X):f\ge0\} \] is its positive cone. We assume that $T$ is positive homogeneous. That is, \[ T(rf)=rT(f) \qquad (r>0,\,f\in C_0^+(X)). \] Under this assumption, we prove that $T$ is represented as a weighted composition operator induced by a homeomorphism from $Y$ onto $X$ and a bounded continuous weight function. Moreover, we show that $T$ extends uniquely to a linear order isomorphism between $C_0(X)$ and $C_0(Y)$.