A Free Probability Analogue of the Wasserstein Metric on the Trace-State Space

Lidskii-type formulae for Dixmier traces

We establish several analogues of the classical Lidskii Theorem for some special classes of singular traces (Dixmier traces and Connes-Dixmier traces) used in noncommutative geometry.

$L^p$-Wasserstein distances on state and quasi-state spaces of $C^*$-algebras

We construct an analogue of the classical $L^p$-Wasserstein distance for the state space of a $C^*$-algebra. Given an abstract Lipschitz gauge on a $C^*$-algebra $\mathcal{A}$ in the sense of Rieffel, one can define the classical $L^p$-Wasserstein distance on the state space of each commutative $C^*$-subalgebra of $\mathcal{A}$. We consider a projective limit of these metric spaces, which appears to be the space of all quasi-linear states, equipped with a distance function. We call this distance the projective $L^p$-Wasserstein distance. It is easy to show, that the state space of a $C^*$-alge…

A quantitative fourth moment theorem in free probability theory

A quantitative "fourth moment theorem" is provided for any self-adjoint element in a homogeneous Wigner chaos: the Wasserstein distance is controlled by the distance from the fourth moment to two. The proof uses the free counterpart of the Stein discrepancy. On the way, the free analogue of the WSH inequality is established.

Weyl's Law and Connes' Trace Theorem for Noncommutative Two Tori

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $\mathbb{T}_θ^2$. We also prove the analogue of Connes' trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order -2 on $\mathbb{T}_θ^2$.

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 p…

完全自由积 $C^*$-代数的迹空间

Trace spaces of full free product $C^*$-algebras

本文研究任意完全自由积的迹空间，证明在无基本障碍时迹空间必为Poulsen单形，并给出群情形下的具体刻画及关键扰动技术。