  This paper studies ways to represent an ordered topological vector space as a
space of continuous functions, extending the classical representation theorems
of Kadison and Schaefer. Particular emphasis is put on the class of semisimple
spaces, consisting of those ordered topological vector spaces that admit an
injective positive representation to a space of continuous functions. We show
that this class forms a natural topological analogue of the regularly ordered
spaces defined by Schaefer in the 1950s, and is characterized by a large number
of equivalent geometric, algebraic, and topological properties.
