Horofunctions and metric compactification of noncompact Hermitian symmetric spaces

Abstract

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 \((V,\Vert \cdot \Vert )\) equipped with a particular Jordan structure, called a \(\textrm{JB}^*\)-triple. We identify the horofunctions in the metric compactification of \((V,\Vert \cdot \Vert )\) and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space \(V^*\). Finally, we show that the exponential map \(\exp _0 :V \longrightarrow D\) at \(0\in D\) extends to a homeomorphism between the metric compactifications of \((V,\Vert \cdot \Vert )\) and \((D,\rho )\), preserving the geometric structure, where \(\rho \) is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of \((V,\Vert \cdot \Vert )\).