Tracial smooth functions of non-commuting variables and the free Wasserstein manifold

We formulate a free probabilistic analog of the Wasserstein manifold on $mathbb{R}^d$ (the formal Riemannian manifold of smooth probability densities on $mathbb{R}^d$), and we use it to study smooth non-commutative transport of measure. The points of the free Wasserstein manifold $mathscr{W}(mathbb{R}^{*d})$ are smooth tracial non-commutative functions $V$ with quadratic growth at $infty$, which correspond to minus the log-density in the classical setting. The space of smooth tracial non-commutative functions used here is a new one whose definition and basic properties we develop in the paper; they are scalar-valued functions of self-adjoint $d$-tuples from arbitrary tracial von Neumann algebras that can be approximated by trace polynomials. The space of non-commutative diffeomorphisms $mathscr{D}(mathbb{R}^{*d})$ acts on $mathscr{W}(mathbb{R}^{*d})$ by transport, and the basic relationship between tangent vectors for $mathscr{D}(mathbb{R}^{*d})$ and tangent vectors for $mathscr{W}(mathbb{R}^{*d})$ is described using the Laplacian $L_V$ associated to $V$ and its pseudo-inverse $Psi_V$ (when defined). Following similar arguments to arXiv:1204.2182, arXiv:1701.00132, and arXiv:1906.10051 in the new setting, we give a rigorous proof for the existence of smooth transport along any path $t mapsto V_t$ when $V$ is sufficiently close $(1/2) sum_j operatorname{tr}(x_j^2)$, as well as smooth triangular transport.