Mathematische Zeitschrift

, Volume 255, Issue 3, pp 579–596

Strong majorization in a free ✱-algebra

  • J. William. Helton
  • Scott McCullough
  • Mihai Putinar


We study, in the spirit of modern real algebra, the interplay between left ideals of the free ∗-algebra \(\mathbb F\) with n generators, and their suitably defined zero sets; and similarly between quadratic submodules of \(\mathbb F\) and their positivity sets.


Positivstellensatz Nullstellensatz Free ∗-algebra 

Mathematics Subject Classification (2000)

46A55 06F25 41A63 


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  1. 1.
    Amitsur S.A. (1957) A generalization of Hilbert’s Nullstellensatz. Proc. Am. Math. Soc. 8, 549–656CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cimpric, J.: A representation theorem for archimedean quadratic modules on ∗-rings, preprint (2004)Google Scholar
  3. 3.
    Cimpric, J.: Maximal quadratic modules on ∗-rings. preprint (2005)Google Scholar
  4. 4.
    Conway, John, B.: A course in operator theory. In: Graduate Studies in Mathematics, vol. 21. American Mathematical Society, Providence, xvi+372 pp (2000)Google Scholar
  5. 5.
    Helton J.W., McCullough S. (2004) A Positivstellensatz for noncommutative polynomials. Trans. Am. Math. Soc. 356, 3721–3737MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Helton J.W., McCullough S., Putinar M. (2005) Non-negative hereditary polynomials in a free ∗-algebra. Math. Zeitschrift 250, 515–522MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Klep, I., Schweighofer, M.: A Nichtnegativstellensatz for polynomials in noncommuting variables. Preprint (2005)Google Scholar
  8. 8.
    Koethe G. (1969) Topological Vector Spaces. I. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  9. 9.
    Krein M.G., Rutman M.A. (1948) Linear operators leaving invariant a cone in a Banach space. (in Russian) Uspehi Mat. Nauk 23, 3–95MathSciNetGoogle Scholar
  10. 10.
    Marshall, M.: Positive Polynomials and Sums of Squares. Instituti Edit. Poligraf. Int. Pisa, Roma (2000)Google Scholar
  11. 11.
    Prestel A., Delzell C.N. (2001) Positive Polynomials. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  12. 12.
    Putinar M. (2006) On hermitian polynomial optimization. Arch. Math. 87, 41–51MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • J. William. Helton
    • 1
  • Scott McCullough
    • 2
  • Mihai Putinar
    • 3
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA
  3. 3.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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