Ideal quantum metrics from fractional Laplacians

Abstract

We develop a novel framework for Monge--Kantorovi\v{c} metrics using Schatten ideals and commutators of fractional Laplacians on Ahlfors regular spaces. Notably, for those metrics we derive closed formulas in terms of spectra of higher-order fractional Laplacians. For our proofs we develop new techniques in noncommutative geometry, in particular a Weyl law and Schatten-class commutators, yielding refined quantum metrics on the space of Borel probability measures. Lastly, our fractional analysis extends to dynamical systems. We showcase this in the setting of expansive algebraic $\mathbb Z^m$-actions and homoclinic $C^*$-algebras of certain hyperbolic dynamical systems. These findings illustrate the versatility of fractional analysis in fractal geometry, dynamical systems and noncommutative geometry.