On Pythagoras' theorem for products of spectral triples

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.

arXiv commentPaper slightly shortened to match the published version; Lett. Math. Phys. 2012

References in corpus (10)

• Gravity and the standard model with neutrino mixing
• Distances in Finite Spaces from Noncommutative Geometry
• Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry
• On the spectral characterization of manifolds
• Product of real spectral triples
• Minimal length in quantum space and integrations of the line element in Noncommutative Geometry
• Bounded and unbounded Fredholm modules for quantum projective spaces
• Line element in quantum gravity: the examples of DSR and noncommutative geometry
• On the Product of Real Spectral Triples
• Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality

Cited by in corpus (11)

• Properties of the Null Distance and Spacetime Convergence
• Exploring the Causal Structures of Almost Commutative Geometries
• On Closed-Form Expressions for the Fisher-Rao Distance
• Length and distance on a quantum space
• On almost commutative Friedmann-Lemaître-Robertson-Walker geometries
• Spectral distances on doubled Moyal plane using Dirac eigen-spinors
• Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality
• Pythagoras Theorem in Noncommutative Geometry
• Connes spectral distance and nonlocality of generalized noncommutative phase spaces
• Connes spectral distances, quantum discord and coherence of qubits
• Connes distance of $2D$ harmonic oscillators in quantum phase space