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Zap. Nauchn. Sem. POMI, 2013, Volume 411, Pages 85–102 (Mi znsl5633)  

This article is cited in 1 scientific paper (total in 1 paper)

Towards a Monge–Kantorovich metric in noncommutative geometry

P. Martinettiab

a Università di Napoli Federico II, I-00185
b CMTP & Dipartimento di Matematica, Università di Roma Tor Vergata, I-00133

Abstract: We investigate whether the identification between Connes' spectral distance in noncommutative geometry and the Monge–Kantorovich distance of order 1 in the theory of optimal transport – that has been pointed out by Rieffel in the commutative case – still makes sense in a noncommutative framework. To this aim, given a spectral triple $(\mathcal A,\mathcal H, D)$ with noncommutative $\mathcal A$, we introduce a “Monge–Kantorovich”-like distance $W_D$ on the space of states of $\mathcal A$, taking as a cost function the spectral distance $d_D$ between pure states. We show in full generality that $d_D\leq W_D$, and exhibit several examples where the equality actually holds true, in particular on the unit two-ball viewed as the state space of $M_2(\mathbb C)$. We also discuss $W_D$ in a two-sheet model (product of a manifold by $\mathbb C^2$), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.

Key words and phrases: Connes distance, spectral triple, state space, Wasserstein distance.

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English version:
Journal of Mathematical Sciences (New York), 2014, 196:2, 165–174

Bibliographic databases:

UDC: 517.972+514.7
Received: 28.02.2013
Language:

Citation: P. Martinetti, “Towards a Monge–Kantorovich metric in noncommutative geometry”, Representation theory, dynamical systems, combinatorial methods. Part XXII, Zap. Nauchn. Sem. POMI, 411, POMI, St. Petersburg, 2013, 85–102; J. Math. Sci. (N. Y.), 196:2 (2014), 165–174

Citation in format AMSBIB
\Bibitem{Mar13}
\by P.~Martinetti
\paper Towards a~Monge--Kantorovich metric in noncommutative geometry
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXII
\serial Zap. Nauchn. Sem. POMI
\yr 2013
\vol 411
\pages 85--102
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5633}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3048270}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2014
\vol 196
\issue 2
\pages 165--174
\crossref{https://doi.org/10.1007/s10958-013-1648-3}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84897035386}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. P. Martinetti, “Connes distance and optimal transport”, Non-Regular Spacetime Geometry, Journal of Physics Conference Series, 968, eds. P. Chrusciel, J. Grant, M. Kunzinger, E. Minguzzi, IOP Publishing Ltd, 2018, 012007  crossref  isi  scopus
  • Записки научных семинаров ПОМИ
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