# Journal of Operator Theory

Volume 88, Issue 2, Fall 2022 pp. 289-308.

Quadratic Wasserstein metrics for von Neumann algebras via transport plans

**Authors**:
Rocco Duvenhage

**Author institution:**Department of Physics,
University of Pretoria,
Pretoria 0002,
South Africa

**Summary: **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 suitableformulation 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.

**DOI: **http://dx.doi.org/10.7900/jot.2021feb18.2317

**Keywords: **Wasserstein metrics, states, von Neumann algebras, bimodules, transport plans, couplings

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