Adjoining an Order Unit to a Matrix Ordered Space

Adjoining an Order Unit to a Matrix Ordered Space 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Adjoining an Order Unit to a Matrix Ordered Space

Positivity , Volume 9 (2) – Dec 2, 2003

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Publisher
Springer Journals
Copyright
Copyright © 2005 by Springer
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-003-2778-5
Publisher site
See Article on Publisher Site

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.

Journal

PositivitySpringer Journals

Published: Dec 2, 2003

References

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