Horofunctions and metric compactification of noncompact Hermitian symmetric spaces

Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit ball D of a Banach space \((V,\Vert \cdot \Vert )\) $(V,\Vert ·\Vert )$ equipped with a particular Jordan structure, called a \(\textrm{JB}^*\) ${\text{JB}}^{\ast }$ -triple. We identify the horofunctions in the metric compactification of \((V,\Vert \cdot \Vert )\) $(V,\Vert ·\Vert )$ and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space \(V^*\) $V^{\ast }$ . Finally, we show that the exponential map \(\exp _0 :V \longrightarrow D\) ${exp}_0:V⟶D$ at \(0\in D\) $0\in D$ extends to a homeomorphism between the metric compactifications of \((V,\Vert \cdot \Vert )\) $(V,\Vert ·\Vert )$ and \((D,\rho )\) $(D,\rho )$ , preserving the geometric structure, where \(\rho \) $\rho$ is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of \((V,\Vert \cdot \Vert )\) $(V,\Vert ·\Vert )$ .