Tracially lyriform $\mathrm{C}^*$-algebras

TL;DR

This paper introduces tracially lyriform $ ext{C}^*$-algebras, extending quantum metric structures to include stronger topologies and classical Wasserstein metrics, with applications to noncommutative geometry and classification of $ ext{C}^*$-algebras.

Contribution

It develops a new category of tracially lyriform $ ext{C}^*$-algebras that generalizes quantum metric spaces and incorporates classical Wasserstein metrics, enhancing the framework for classifiable $ ext{C}^*$-algebras.

Findings

Constructed examples from fractals and Alexandrov spaces.

Established behavior under inductive limits.

Analyzed geometry and statistics of embedding spaces.

Abstract

Quantum metric Choquet simplices are special kinds of compact quantum metric spaces designed for distance measurement in and around the category of stably finite Elliott-classifiable $\mathrm{C}^*$-algebras. The primary objective of this article is to introduce versions of these structures for which the associated tracial metrics need not be induced by Lipschitz seminorms and may induce strictly stronger topologies than the weak$^*$-topology. The resulting category of 'tracially lyriform $\mathrm{C}^*$-algebras' behaves well with respect to sequential inductive limits and accommodates the full family of classical $p$-Wasserstein metrics on probability spaces, including $p=\infty$. Examples of projectionless, classifiable tracial Wasserstein spaces are built as noncommutative spaces of observables of certain compact length spaces, including: fractals like the Sierpi\'nski gasket, the Sierpi\'nski carpet and the Menger sponge; finite-dimensional Alexandrov spaces with two-sided curvature bounds; and metric spaces like simplicial spheres and balls that are Lipschitz equivalent to Riemannian. These simplicial structures are used as building blocks that furnish arbitrary simple inductive limits of prime dimension drop algebras with tracial lyriform structure. Appealing to optimal transport theory, we study the geometry and statistics of the spaces of embeddings of these building blocks and their limits into suitable classifiable $\mathrm{C}^*$-algebras like the Jiang-Su algebra $\mathcal{Z}$ or the universal UHF algebra $\mathcal{Q}$.