Skip to main content
SpringerLink
Log in

A p-theory of ordered normed spaces

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We propose a pair of axioms (O.p.1) and (O.p.2) for 1 ≤ p ≤ ∞ and initiate a study of a (matrix) ordered space with a (matrix) norm, in which the (matrix) norm is related to the (matrix) order. We call such a space a (matricially) order smooth p-normed space. The advantage of studying these spaces over L p-matricially Riesz normed spaces is that every matricially order smooth ∞-normed space can be order embedded in some C*-algebra. We also study the adjoining of an order unit to a (matricially) order smooth ∞-normed space. As a consequence, we sharpen Arveson’s extension theorem of completely positive maps. Another combination of these axioms yields an order theoretic characterization of the set of real numbers amongst ordered normed linear spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arveson W.B.: Subalgebras of C*-algebras. Acta Math. 123, 141–224 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blecher, D.P., Neal, M.: Open partial isometries and positivity in operator spaces (preprint, 2006)

  3. Choi M.D., Effros E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jameson, G.J.O.: Ordered linear spaces. Lecture Notes in Mathematics, vol. 141. Springer, New York (1970)

  5. Karn A.K.: Adjoining an order unit in a matrix ordered space. Positivity 9, 207–223 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Karn A.K.: Corrigendum to the paper “Adjoining an order unit in a matrix ordered space”. Positivity 11, 369–374 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Karn, A.K.: Order embedding of matrix ordered spaces. (Communicated for publication)

  8. Karn A.K., Vasudevan R.: Matrix norms in matrix ordered spaces. Glasnik Mathematic̆i 32(52), 87–97 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Karn A.K., Vasudevan R.: Approximate matrix order unit spaces. Yokohama Math. J. 44, 73–91 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Karn A.K., Vasudevan R.: Matrix duality for matrix ordered spaces. Yokohama Math. J. 45, 1–18 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Karn A.K., Vasudevan R.: Characterizations of matricially Riesz normed spaces. Yokohama Math. J. 47, 143–153 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Ruan Z.J.: Subspaces of C*-algebras. J. Funct. Anal. 76, 217–230 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Schreiner W.J.: Matrix regular operator spaces. J. Funct. Anal. 152, 136–175 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Werner W.: Subspaces of L(H) that are *-invariant. J. Funct. Anal. 193, 207–223 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Karam Pura, New Delhi, 110 015, India

    Anil K. Karn

Authors
  1. Anil K. Karn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anil K. Karn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karn, A.K. A p-theory of ordered normed spaces. Positivity 14, 441–458 (2010). https://doi.org/10.1007/s11117-009-0029-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-009-0029-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation