Quantum metric Choquet simplices
Jacelon, Bhishan (Czech Academy of Sciences. Institute of Mathematics)

Date:	2026
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 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 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 C*algebra B. As for examples, we recall the construction of classifiable C*-algebraic quantum metric Bauer simplices that function as noncommutative spaces of observables of compact connected metric spaces (X; p). We also explain how to build non-Bauer examples by forming 'tracial quantum crossed products' associated with topological dynamical systems on (X; p), and we use classification to show 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 with respect to Rieffel's quantum Gromov-Hausdor distance. As an example, we present a field of deformed tracial rotation algebras whose fibres are continuous with respect to a quasimetric that we call the quantum intertwining gap.
Note:	This work is dedicated to the former and future members of Prague’s NCG&T group, to our family and friends, to our fathers and forefathers. My research was supported by the Czech Science Foundation (GACR) project 22-07833K ˇ and the Institute of Mathematics of the Czech Academy of Sciences (RVO: 67985840). I am grateful to Hanfeng Li and Stuart White for helpful discussions at the Fields Institute during the Thematic Program on Operator Algebras and Applications in the autumn of 2023, and to the members of the Operator Algebra group at the Southern University of Denmark for their kind hospitality and useful feedback during my visit to Odense in February 2024.
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Quantum metric spaces ; Choquet simplices ; C*-dynamical systems ; C*-classification
Published in:	Publicacions matemàtiques, Vol. 70, Num. 2 (2026) , p. 339-377 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/1500000000000137
DOI: 10.5565/PUBLMAT7022603

39 p, 1.2 MB

The record appears in these collections:
Articles > Published articles > Publicacions matemàtiques
Articles > Research articles