Letters in Mathematical Physics

, Volume 103, Issue 5, pp 469–492 | Cite as

On Pythagoras Theorem for Products of Spectral Triples

Article

Abstract

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes’ distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K-homology.

Mathematics Subject Classification (2010)

58B34 46L87 

Keywords

noncommutative geometry spectral triples spectral distance Pythagoras theorem 

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References

  1. 1.
    Baaj S., Julg P.: Théorie bivariante de Kasparov et opérateurs non bornés dans les C *-modules hilbertiens. C. R. Acad. S. Paris 296, 875–878 (1983)MathSciNetMATHGoogle Scholar
  2. 2.
    Chamseddine A.H., Connes A., Marcolli M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)MathSciNetMATHGoogle Scholar
  3. 3.
    Connes A.: Noncommutative differential geometry. Inst. Hautes Etudes Sci. Publ. Math. 62, 257–360 (1985)MathSciNetGoogle Scholar
  4. 4.
    Connes A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergod. Theory Dyn. Syst. 9, 207–220 (1989)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Connes A.: Noncommutative Geometry. Academic Press, New York (1994)MATHGoogle Scholar
  6. 6.
    Connes A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Connes, A.: On the spectral characterization of manifolds. arXiv:0810.2088v1 [math.OA]Google Scholar
  8. 8.
    Connes, A.: Variations sur le thème spectral. Summary of Collège de France lectures (2007). http://www.college-de-france.fr/media/alain-connes/UPL53971_2.pdf; see also http://noncommutativegeometry.blogspot.com/search?q=harmonic+mean
  9. 9.
    D’Andrea F., Landi G.: Bounded and unbounded Fredholm modules for quantum projective spaces. J. K-Theory 6, 231–240 (2010)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    D’Andrea F., Martinetti P.: A view on transport theory from noncommutative geometry. SIGMA 6, 057 (2010)MathSciNetGoogle Scholar
  11. 11.
    Dabrowski L., Dossena G.: Product of real spectral triples. Int. J. Geom. Methods Mod. Phys. 8, 1833–1848 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gilkey, P.B.: Invariance theory, the heat equation and the Atiyah-Singer index theorem. Publish or Perish Inc (1984)Google Scholar
  13. 13.
    Golub G.H., van Loan C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  14. 14.
    Gracia-Bondía J.M., Várilly J.C., Figueroa H.: Elements of Noncommutative Geometry. Birkhäuser, Basel (2001)MATHGoogle Scholar
  15. 15.
    Grothendieck A.: Produits Tensoriels Topologiques et Espaces Nucléaires. Memoirs of the AMS, vol. 16. AMS, Providence (1979)Google Scholar
  16. 16.
    Higson N., Roe J.: Analytic K-Homology. Oxford University Press, NY (2000)MATHGoogle Scholar
  17. 17.
    Iochum B., Krajewski T., Martinetti P.: Distances in finite spaces from noncommutative geometry. J. Geom. Phys. 31, 100–125 (2001)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Kadison R.V., Ringrose J.R.: Fundamentals of the Theory of Operator Algebras, vol. II. Advanced Theory, Vol. IV. Academic Press, New York (1986)Google Scholar
  19. 19.
    Martinetti P., Mercati F., Tomassini L.: Minimal length in quantum space and integrations of the line element in noncommutative geometry. Rev. Math. Phys. 24, 1250010–1250045 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Martinetti, P., Tomassini, L.: Noncommutative geometry of the Moyal plane: translation isometries and spectral distance between coherent states. arXiv:1110.6164v1 [math-ph]Google Scholar
  21. 21.
    Martinetti P., Wulkenhaar R.: Discrete Kaluza–Klein from scalar fluctuations in noncommutative geometry. J. Math. Phys. 43(1), 182–204 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Martinetti P.: Line element in quantum gravity: the examples of DSR and noncommutative geometry. Int. J. Mod. Phys. A 24(15), 2792–2801 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Rieffel M.A.: Metric on state spaces. Doc. Math. 4, 559–600 (1999)MathSciNetMATHGoogle Scholar
  24. 24.
    Vanhecke F.-J.: On the product of real spectral triples. Lett. Math. Phys. 50, 157–162 (1999)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Villani C.: Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften vol. 338. Springer, Berlin (2009)Google Scholar
  26. 26.
    Wegge-Olsen, N.E.: K-Theory and C *-Algebras. A Friendly Approach. Oxford University Press, Oxford (1993)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Napoli Federico IINaplesItaly
  2. 2.Dipartimento di Matematica e CMTPUniversità di Roma Tor VergataRomeItaly
  3. 3.Dipartimento di FisicaUniversità di Roma SapienzaRomeItaly

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