$L^{2}$-Wasserstein distances of tracial $W^{*}$-algebras and their disintegration problem

We introduce $L^{2}$-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 $L^{2}$-Wasserstein distances induced by decomposed derivations on $C^{*}$-algebras of continuous sections of a $\mathcal{K}(H)$-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 $L^{2}$-Wasserstein distances for unital $C^{*}$-algebras that are Morita equivalent to a commutative unital $C^{*}$-algebra.