Extension of order-preserving maps on a cone

We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone $K_1$ (taking values in another closed cone $K_2$) to the whole of $K_1$. We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous. Maps $f$ defined on the interior of the non-negative cone $K$ in $R^N$, which are both homogeneous of degree 1 and order preserving, are non-expanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that such an extension must have at least one eigenvector in $K – \{0\}$. In the case where the cycle time $\chi(f)$ of the original map does not exist, such eigenvectors must lie in $\partial K – \{0\}$. We conclude with some discussions and applications to operator-valued means. We also extend our results to an ‘intermediate’ situation, which arises in some important application areas, particularly in the construction of diffusions on certain fractals via maps defined on the interior of cones of Dirichlet forms.