Letters in Mathematical Physics

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

On Pythagoras Theorem for Products of Spectral Triples



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 


noncommutative geometry spectral triples spectral distance Pythagoras theorem 


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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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