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Abstract
We examine the problem of extending, in a natural way, orderpreserving 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 infinitedimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semicontinuous. Maps $f$ defined on the interior of the nonnegative cone $K$ in $R^N$, which are both homogeneous of degree 1 and order preserving, are nonexpanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous orderpreserving 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 operatorvalued 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.
Original language  English 

Pages (fromto)  35 
Number of pages  1 
Journal  Proceedings of the Royal Society of Edinburgh: Section A Mathematics 
Volume  133 
Issue number  1 
DOIs  
Publication status  Published  2003 
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Dive into the research topics of 'Extension of orderpreserving maps on a cone'. Together they form a unique fingerprint.Activities
 1 Visiting an external academic institution

Rutgers  The State University of New Jersey, New Brunswick
Andrew Burbanks (Visiting researcher)
1 Jan 2002 → 31 Dec 2003Activity: Visiting an external organisation types › Visiting an external academic institution