A p-theory of ordered normed spaces

We propose a pair of axioms (O.p.1) and (O.p.2) for 1 a parts per thousand currency sign p a parts per thousand currency sign a and initiate a study of a (matrix) ordered space with a (matrix) norm, in which the (matrix) norm is related to the (matrix) order. We call such a space a (matricially) order smooth p-normed space. The advantage of studying these spaces over L (p) -matricially Riesz normed spaces is that every matricially order smooth a-normed space can be order embedded in some C*-algebra. We also study the adjoining of an order unit to a (matricially) order smooth a-normed space. As a consequence, we sharpen Arveson's extension theorem of completely positive maps. Another combination of these axioms yields an order theoretic characterization of the set of real numbers amongst ordered normed linear spaces.