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 dD≤WD, 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.
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
\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:
D'Andrea F., Martinetti P., “A Dual Formula For the Spectral Distance in Noncommutative Geometry”, J. Geom. Phys., 159 (2021), 103920
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
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