Metrics on trace spaces

TL;DR

This paper explores the tracial geometry of certain classifiable C*-algebras, using optimal transport to compute distances between unitary orbits via tracial data and applying Lipschitz structures to statistical properties of endomorphisms.

Contribution

It introduces a framework for calculating distances between *-homomorphisms in classifiable C*-algebras using optimal transport and tracial Lipschitz elements, with applications to statistical features.

Findings

Distance between unitary orbits can be computed via tracial data.

Tracial Lipschitz elements can witness statistical features of endomorphisms.

Application of optimal transport to noncommutative CW complexes.

Abstract

This article continues the investigation of the tracial geometry of classifiable $\mathrm{C}^*$-algebras that have real rank zero and stable rank one. Using the language of optimal transport, we describe several situations in which the distance between unitary orbits of $^*$-homomorphisms into such algebras can be computed in terms of tracial data. The domains we consider are certain (noncommutative) CW complexes, and the measurement is relative to a family of self-adjoint elements that are in a suitable sense tracially Lipschitz. As another application of the utility of this Lipschitz structure, we show how such elements can be repurposed to witness statistical features of endomorphisms in the classifiable category, in particular the tracial version of the (almost-sure) central limit theorem.