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Zapiski Nauchnykh Seminarov POMI, 2013, Volume 411, Pages 85–102 (Mi znsl5633)  

This article is cited in 3 scientific papers (total in 3 papers)

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
Full-text PDF (458 kB) Citations (3)
References:
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 (A,H,D) with noncommutative A, we introduce a “Monge–Kantorovich”-like distance WD on the space of states of A, taking as a cost function the spectral distance dD between pure states. We show in full generality that dDWD, and exhibit several examples where the equality 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 (product of a manifold by C2), 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.
Received: 28.02.2013
English version:
Journal of Mathematical Sciences (New York), 2014, Volume 196, Issue 2, Pages 165–174
DOI: https://doi.org/10.1007/s10958-013-1648-3
Bibliographic databases:
Document Type: Article
UDC: 517.972+514.7
Language: English
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://mathscinet.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{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84897035386}
Linking options:
  • https://www.mathnet.ru/eng/znsl5633
  • https://www.mathnet.ru/eng/znsl/v411/p85
  • This publication is cited in the following 3 articles:
    1. D'Andrea F., Martinetti P., “A Dual Formula For the Spectral Distance in Noncommutative Geometry”, J. Geom. Phys., 159 (2021), 103920  crossref  mathscinet  zmath  isi  scopus
    2. 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  mathscinet  isi  scopus
    3. Francesco D'Andrea, Fedele Lizzi, Pierre Martinetti, “Spectral geometry with a cut-off: Topological and metric aspects”, Journal of Geometry and Physics, 82 (2014), 18  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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