Compactification structure and conformal compressions of symmetric cones

In this paper we show that the boundary of a symmetric cone Ω in the standard real conformal compactification M of its containing euclidean Jordan algebra V has the structure of a double cone, with the points at infinity forming one of the cones. We further show that Ω̄^M admits a natural partial order extending that of Ω. Each element of the compression semigroup for Ω is shown to act in an order-preserving way on Ω̄^M and carries it into an order interval contained in Ω̄^M.