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Adjoining an Order Unit to a Matrix Ordered Space

Authors
  • Karn, Anil K.1
  • 1 University of Delhi, Department of Mathematics, Deen Dayal Upadhyaya College, Karam Pura, New Delhi, 110 015, India , Karam Pura
Type
Published Article
Journal
Positivity
Publisher
Kluwer Academic Publishers
Publication Date
Jun 01, 2005
Volume
9
Issue
2
Pages
207–223
Identifiers
DOI: 10.1007/s11117-003-2778-5
Source
Springer Nature
Keywords
License
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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.

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