On Pythagoras Theorem for Products of Spectral Triples |
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Authors: | Francesco D’Andrea Pierre Martinetti |
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Affiliation: | 1. Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Naples, Italy 2. Dipartimento di Matematica e CMTP, Università di Roma Tor Vergata, Rome, Italy 3. Dipartimento di Fisica, Università di Roma Sapienza, Rome, Italy
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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. |
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