L²-Wasserstein distances of tracial W^*-algebras and their disintegration problem

2018/06/04 by Hornshaw, David F. · 1 citation
#46L53 (Primary) 46L87 (Secondary) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1806.01073

Abstract

We introduce L²-Wasserstein distances on densities of tracial W^*-algebras based on a Benamou-Brenier formulation, replacing multiplication by densities with multiplication operators arising as the logarithmic mean under a functional calculus. Furthermore, we concern ourselves with L²-Wasserstein distances induced by decomposed derivations on C^*-algebras of continuous sections of a K(H)-bundle vanishing at infinity. We prove a distintegration theorem for such distances, introduce mean entropic curvature bounds in case H is finite-dimensional and show control of these by the essential infimum of the entropic curvature bounds on the fibres. To conclude, we give sufficient conditions for disintegrating arbitrary L²-Wasserstein distances for unital C^*-algebras that are Morita equivalent to a commutative unital C^*-algebra.

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