Roller boundaries for median spaces and algebras

Abstract

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of $CAT\left(0\right)$ cube complexes. Examples of median spaces with compact intervals include all finite-rank median spaces and all proper median spaces of infinite rank. Our methods also apply to general median algebras, where we recover the zero-completions of Bandelt and Meletiou (1993). Along the way, we prove various properties of halfspaces in finite-rank median spaces and a duality result for locally convex median spaces.