On Pythagoras' theorem for products of spectral triples

TL;DR

This paper explores a noncommutative geometry version of Pythagoras' theorem, extending it from pure states to general states and arbitrary spectral triples, resulting in inequalities rather than equalities.

Contribution

It generalizes Pythagoras' theorem in noncommutative geometry to non-pure states and non-commutative spectral triples, establishing optimal inequalities under unitality assumptions.

Findings

Pythagoras theorem becomes inequalities in noncommutative setting

Inequalities are proven to be optimal

Counter-examples show limitations without unitality

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 provide non-unital counter-examples inspired by K-homology.