Quadratic Wasserstein metrics for von Neumann algebras via transport plans

Abstract

We show how one can obtain a class of quadratic Wasserstein metrics, that is to say, Wasserstein metrics of order 2, on the set of faithful normal states of a von Neumann algebra $A$, via transport plans, rather than through a dynamical approach. Two key points to make this work, are a suitable formulation of the cost of transport arising from Tomita-Takesaki theory and relative tensor products of bimodules (or correspondences in the sense of Connes). The triangle inequality, symmetry and $W_2(\mu ,\mu )=0$ all work quite generally, but to show that $W_2(\mu ,\nu )=0$ implies $\mu =\nu$, we need to assume that $A$ is finitely generated.

v1: 20 pages. v2: typos fixed, minor corrections, added remarks and references, to appear in Journal of Operator Theory, 22 pages. v3: corrected typos in Prop 4.3's proof. v4: minor corrections and two references added

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