Quantum metric Choquet simplices

Abstract

Precipitating a notion emerging from recent research, we formalise the study of a special class of compact quantum metric spaces. Abstractly, the additional requirement we impose on the underlying order unit spaces is the Riesz interpolation property. In practice, this means that a `quantum metric Choquet simplex' arises as a unital $\mathrm{C}^*$-algebra $A$ whose trace space is equipped with a metric inducing the $w^*$-topology, such that tracially Lipschitz elements are dense in $A$. This added structure is designed for measuring distances in and around the category of stably finite classifiable $\mathrm{C}^*$-algebras, and in particular for witnessing metric and statistical properties of the space of (approximate unitary equivalence classes of) unital embeddings of $A$ into a stably finite classifiable $\mathrm{C}^*$-algebra $B$. Our reference frame for this measurement is a certain compact `nucleus' of $A$ provided by its quantum metric structure. As for the richness of the metric space of isometric isomorphism classes of classifiable $\mathrm{C}^*$-algebraic quantum metric Choquet simplices (equipped with Rieffel's quantum Gromov--Hausdorff distance), we show how to construct examples starting from Bauer simplices associated to compact metric spaces. We also explain how to build non-Bauer examples by forming `quantum crossed products' associated to dynamical systems on the tracial boundary. Further, we observe that continuous fields of quantum spaces are obtained by continuously varying either the dynamics or the metric. In the case of deformed isometric actions, we show that equivariant Gromov--Hausdorff continuity implies fibrewise continuity of the quantum structures. As an example, we present a field of deformed quantum rotation algebras whose fibres are continuous with respect to a quasimetric called the quantum intertwining gap.