Adjoining an Order Unit to a Matrix Ordered Space

Abstract

We prove that an order unit can be adjoined to every L ^∞-matricially Riesz normed space. We introduce a notion of strong subspaces. The matrix order unit space obtained by adjoining an order unit to an L ^∞-matrically Riesz normed space is unique in the sense that the former is a strong L ^∞-matricially Riesz normed ideal of the later with codimension one. As an application of this result we extend Arveson’s extension theorem to L ^∞-matircially Riesz normed spaces. As another application of the above adjoining we generalize Wittstock’s decomposition of completely bounded maps into completely positive maps on C ^*-algebras to L ^∞-matricially Riesz normed spaces. We obtain sharper results in the case of approximate matrix order unit spaces.