On Pythagoras' theorem for products of spectral triples

Details On Pythagoras' theorem for products of spectral triples On Pythagoras' theorem for products of spectral triples

2012-03-14

arxiv.org/abs/1203.3184v1

Physics

Mathematical Physics

Scientific paper

We discuss a version of Pythagoras' theorem in noncommutative geometry. Starting from a formulation of the theorem in terms of Connes' distance, between pure states, in the product of commutative spectral triples, we show that for non-pure states it is replaced by some Pythagoras inequalities. We prove the latter in full generality, that is for the product of arbitrary (i.e. non-necessarily commutative) unital spectral triples. Moreover we show that the inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.

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