Geodesic
Towards a Monge–Kantorovich metric in noncommutative geometry
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 85-102 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Towards {a~Monge{\textendash}Kantorovich} metric in noncommutative geometry},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a4/}
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P. Martinetti. Towards a Monge–Kantorovich metric in noncommutative geometry. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 85-102. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a4/

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