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.
Positivity – Springer Journals
Published: Dec 2, 2003
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