Horofunction compactifications of symmetric cones under Finsler distances

Abstract

In this paper we consider symmetric cones as symmetric spaces equipped with invariant Finsler distances, namely the Thompson distance and the Hilbert distance. We establish a correspondence between the horofunction compactification of a symmetric cone $A_+^\circ$ under these invariant Finsler distances and the horofunction compactification of the normed space in the tangent bundle. More precisely, for the Thompson distance on $A^\circ_+$ we show that the exponential map extends as a homeomorphism between the horofunction compactification of the normed space in the tangent bundle, which is a JB-algebra, and the horofunction compactification of $A_+^\circ$. We give a complete characteristation of the Thompson distance horofunctions and provide an explicit extension of the exponential map. Analogues results are established for the Hilbert distance on the projective cone $PA_+^\circ$. The analysis yields a geometric description of the horofunction compactifications of these spaces in terms of the facial structure of the closed unit ball of the dual norm of the norm in the tangent space.