2012/03/31 by Francesco D’Andrea, Francesco D'Andrea, Pierre Martinetti · 2 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #math-ph #math.MP #math.OA #msc:46L87 #msc:58B34
paper · pdf · doi:10.1007/s11005-012-0598-x
Paper slightly shortened to match the published version; Lett. Math. Phys. 2012
openalex publication_date 2012/11/30 · arxiv created 2012/12/05 · arxiv updated 2012/12/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29
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.