Extension of order-preserving maps on a cone

We examine the problem of extending, in a natural way, order-preserving maps that are de ̄ned on the interior of a closed cone K1 (taking values in another closed cone K2 ) to the whole of K1 . We give conditions, in considerable generality (for cones in both ̄niteand in ̄nite-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 de ̄ned on the interior of the non-negative cone K in R , 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 ¡f0g. In the case where the cycle time À (f ) of the original map does not exist, such eigenvectors must lie in @K ¡f0g. We conclude with some discussions and applications to operator-valued means. We also extend our results to an ìntermediate’ situation, which arises in some important application areas, particularly in the construction of di® usions on certain fractals via maps de ̄ned on the interior of cones of Dirichlet forms.