Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas

  • Konrad Schmüdgen
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 149)

Abstract

We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensätze) from real algebraic geometry can be generalized to noncommutative *-algebras. A version of Stengle's Positivstellensatz for n X n matrices of real polynomials is proved.

Key words

Noncommutative real algebraic geometry quadratic module, sum of squares *-representation positivity positive semidefinite matrices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.S. BIRMAN AND M.Z. SOLOMJAK, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987.Google Scholar
  2. J. BOCHNAK, M. COSTE, AND M.-F. RoY, Real Algebraic Geometry, SpringerVerlag, Berlin, 1998.Google Scholar
  3. J. CIMPRIC, Maximal quadratic modules on *-rings, Algebr. Represent. Theory (2007), DOl  10.1007/S10468-007-9076-z.
  4. J. CIMPRIC, A representation theorem for quadratic modules on «-rlngs, 2005, to appear in Canadian Mathematical Bulletin.Google Scholar
  5. D.Z. DJOKOVIC, Hermitean matrices over polynomial rings, J. Algebra 43 (1976), pp. 359-374.Google Scholar
  6. J. FRIEDRlCH AND K. SCHMUDGEN, n-Positivity of unbounded «-represenieilons, Math. Nachr. 141 (1989), pp. 233~250.Google Scholar
  7. F.R. GANTMACHER, Matrizentbeorie, DVW, Berlin, 1986.Google Scholar
  8. D. GONDARD AND P. RIBENBOIM, Le 17e probleme de Hilbert pour les matrices, Bull. Sci. Math. 98 (1974), pp. 49-56.Google Scholar
  9. [9} J.W. HELTON, Positive noncommutative polynomials are sums of squares, Ann. Math. 156 (2002), pp. 675-694.Google Scholar
  10. J.W. HELTON AND S. MCCULLOUGH, A Positivstellensatz for non-commutative polynomials, Trans. Amer. Math. Soc. 356 (2004), pp. 3721-3737.Google Scholar
  11. J.W. HELTON, S. MCCULLOUGH, AND M. PUTINAR, A non-commutative Positivstellensatz on isometrics, J. Reine Angew. Math. 568 (2004), pp. 71-80.Google Scholar
  12. T. JACOBI, A representation theorem for certain partially ordered commutaUve rings, Math. Z. 237 (2001), pp. 259-273.Google Scholar
  13. T. JACOBI AND A. PRESTEL, Distingiushed representations of strictly positive polynomials, J. reine angew. Math. 532 (2001), 223-235.Google Scholar
  14. [14} G. JAMESON, Ordered Linear Spaces, Lecture Notes in Math. No. 141, SpingerVerlag, Berlin, 1970.Google Scholar
  15. A. KLIMYIK AND K. SCHMUDGEN, Quantum Groups and Their Representations, Springer-Verlag, Berlin, 1997.Google Scholar
  16. M. KNEBUSCH AND C. SCHEIDERER, Einfiihrung in die reelle Algebra, ViewegVerlag, Braunschweig, 1989.Google Scholar
  17. V.M. MANUILOV AND E.V. TROITSKY, Hilbert C*-Modules, Amer. Math. Soc., Transl. Math. Monographs 226, 2005.Google Scholar
  18. M. MARSHALL, Positive Polynomials and sums of Squares, Univ. Pisa, Dipart. Mat. Istituti Editoriali e Poligrafici Internaz., 2000.Google Scholar
  19. M. MARSHALL, Extending the Archimedean Positivstellensatz to the non-compact case, Can. Math. Bull. 14(2001), 223~230.Google Scholar
  20. V. OSTROVSKYJ AND Yu. SAMOILENKO, Introduction to the Theory of Finitely Presented *-Algebras, Harwood Acad. Publ., 1999.Google Scholar
  21. V. POWERS AND C. SCHEIDERER, The moment problem for non-compact semialgebraic sets, Adv. Geom.l (2001), pp. 71-88.Google Scholar
  22. A. PRESTEL AND CH.N. DELZELL, Positive Polynomials, Spinger-Verlag, Berlin, 200l.Google Scholar
  23. C. PROCESI AND M. SCHACHER, A Non-Commutative Real Nullstellensatz and Hilbert's 17th Problem, Ann Math. 104 (1976), pp. 395-406.Google Scholar
  24. M. PUTINAR, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42, pp. 969-984.Google Scholar
  25. [25} M. PUTINAR AND F. VASILESCU, Solving moment problems by dimension extension, Ann. Math. 149, pp. 1087-1107.Google Scholar
  26. [26J H. SCHAFER, Topological Vector Spaces, Springer-Verlag, Berlin, 1972.Google Scholar
  27. [27J K. SCHMUDGEN, Graded and filtrated topological *-algebras II. The closure of the positive cone, Rev. Roum. Math. Pures et Appl. 29 (1984), pp. 89-96.Google Scholar
  28. {28J K. SCHMUDGEN, Unbounded Operator Algebras and Representation Theory, Birkhauser-VeTlag, Basel, 1990.Google Scholar
  29. K. SCHMUDGEN, The K -moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), pp. 203-206.Google Scholar
  30. K. SCHMUDGEN, A strict Posltivstellensetz for the Weyl algebra, Math. Ann. 331 (2005), pp. 779-794.Google Scholar
  31. K. SCHMUDGEN, A strict Positivstellenseiz for enveloping algebras, Math. Z. 254 (2006), pp. 641--653. Erratum; DOl  10.1007/s00209-007-0295-0.
  32. M. SCHWEIGHOFER, Iterated rings of bounded elements and generalizations of Scbmiidgen's theorem, J. Reine Angew. Math. 554 (2003), pp. 19-45.Google Scholar
  33. M. SCHWEIGHOFER, An algorithmic approach to Schmiidgen's Positivstellensatz, J. Pure Appl. Algebra 166 (2002), pp. 307-309.Google Scholar
  34. T. WORMANN, Strict positive Polynome in der semielgebreischeti Geometrie, Dissertation, Universitjit Dortmund, 1998.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Konrad Schmüdgen
    • 1
  1. 1.Fakultät für Mathematik und InformatikUniversität LeipzigLeipzigGermany

Personalised recommendations