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Positivity

, Volume 22, Issue 2, pp 597–607 | Cite as

Hilbert \(C^*\)-modules as a subcategory of operator systems and injectivity

  • Mohammad B. Asadi
  • Reza Behmani
  • Ali R. Medghalchi
  • Hamed Nikpey
Article
  • 63 Downloads

Abstract

In this paper, we study a category whose objects are Hilbert \(C^*\)-modules and whose morphisms are completely semi-\(\phi \)-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-\(\phi \)-maps on Hilbert \(C^*\)-modules, leading to an analog of the Arveson’s extension theorem for completely semi-\(\phi \)-maps (in contrast with \(\phi \)-maps). This theorem together with previous results suggest that the completely semi-\(\phi \)-maps are proper generalizations of the completely positive maps.

Keywords

Hilbert \(C^*\)-modules Extension theorems Injective objects Completely positive maps Completely semi-\(\phi \)-maps 

Mathematics Subject Classification

Primary 46L08 Secondary 46L07 

Notes

Acknowledgements

The research of the first author was in part supported by a Grant from IPM (No. 94470046).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  3. 3.Department of MathematicsKharazmi UniversityTehranIran
  4. 4.Department of MathematicsShahid Rajaee Teacher Training UniversityTehranIran

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