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.