Isometric representations of calibrated ordered spaces on $C(X)$

The problem of characterizing normed ordered spaces which admit a representation in the algebraic, order and norm sense as a subspace of $C(X)$, the space of all continuous functions on a compact Hausdorff space is a classical problem that has been considered by many authors. In this article we consider the more general case of calibrated ordered spaces, that is, ordered spaces with a specified family of seminorms generating its topology. For such spaces equivalent conditions on representability as a subspace of $C(X)$ for some locally compact Hausdorff space $X$, in the algebraic, order and seminorm sense are stated and proved. Some characterizations appear to be new even in the normed case. A recent result on isometric representations of locally ordered spaces fall under the results in this paper with more general statements. As an application of the main theorems, we state and prove a characterization of norm additivity property of two positive functionals.