Journal of Mathematical Sciences

, Volume 196, Issue 2, pp 165–174

Towards a Monge − Kantorovich Metric in Noncommutative Geometry

Article

DOI: 10.1007/s10958-013-1648-3

Cite this article as:
Martinetti, P. J Math Sci (2014) 196: 165. doi:10.1007/s10958-013-1648-3

We investigate whether the identification between Connes’s spectral distance in noncommutative geometry and the Monge–Kantorovich distance of order 1 in the theory of optimal transport which has been pointed out by Rieffel in the commutative case still makes sense in a noncommutative framework. To this aim, given a spectral triple (\( \mathcal{A} \), \( \mathcal{H} \), \( \mathcal{D} \)) with noncommutative \( \mathcal{A} \), we introduce a "Monge–Kantorovich"-like distance WD on the space of states of \( \mathcal{A} \), taking as a cost function the spectral distance dD between pure states. We show in full generality that dD ≤ WD, and exhibit several examples where thee quality actually holds true, in particular, on the unit two-ball viewed as the state space of M2(ℂ). We also discuss WD in a two-sheet model (the product of a manifold and ℂ2), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CMTP & Dipartimento di MatematiaUniversità di Roma Tor Vergata, Università di Napoli Federico IIRomeItaly
  2. 2.CMTP & Dipartimento di MatematiaUniversità di Napoli Federico IINapoliItaly

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