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Quantum metric Choquet simplices

Bhishan Jacelon

TL;DR

This work develops a robust framework for quantum metric geometry in the setting of classifiable $ m C^*$-algebras by employing the Cuntz–Pedersen quotient $A^q$ and the Riesz interpolation property to define quantum metric Choquet simplices (QMCS) and quantum metric Bauer simplices (QMBS). It constructs canonical noncommutative observables for metric spaces via Bauer simplex models $( rak A_X,L_ ho)$ and shows how tracial quantum crossed products $A_X times_{oldsymbol{eta}_h}bN$ inherit QMCS structure, especially when actions have finite Rokhlin dimension; this allows one to study the metric and statistical properties of embeddings into stably finite classifiable algebras. The paper introduces a quantum Gromov–Hausdorff distance dist$_q$ and the quantum intertwining gap $oldsymbol{eta}_q$, proving completeness for QMBS and establishing a topological equivalence between these invariants; it also develops continuous-field techniques to analyze metric and dynamical deformations, including random and dynamical systems, and demonstrates that metrisableChoquet simplices arise as trace spaces of dynamical QMCS. Collectively, these results provide a bridge between noncommutative geometry, classification theory, and dynamical systems, enabling precise control over how quantum metric structures vary under deformations and embeddings. The framework is illustrated through examples such as tracial rotation algebras and projectionless models, highlighting the interplay between trace geometry, $K$-theory, and continuous variation of dynamics and metrics.

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 $B$. As for examples, we recall the construction of classifiable $\mathrm{C}^*$-algebraic quantum metric Bauer simplices that function as noncommutative spaces of observables of compact connected metric spaces $(X,ρ)$. We also explain how to build non-Bauer examples by forming 'tracial quantum crossed products' associated with topological dynamical systems on $(X,ρ)$, 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-Hausdorff 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.

Quantum metric Choquet simplices

TL;DR

This work develops a robust framework for quantum metric geometry in the setting of classifiable -algebras by employing the Cuntz–Pedersen quotient and the Riesz interpolation property to define quantum metric Choquet simplices (QMCS) and quantum metric Bauer simplices (QMBS). It constructs canonical noncommutative observables for metric spaces via Bauer simplex models and shows how tracial quantum crossed products inherit QMCS structure, especially when actions have finite Rokhlin dimension; this allows one to study the metric and statistical properties of embeddings into stably finite classifiable algebras. The paper introduces a quantum Gromov–Hausdorff distance dist and the quantum intertwining gap , proving completeness for QMBS and establishing a topological equivalence between these invariants; it also develops continuous-field techniques to analyze metric and dynamical deformations, including random and dynamical systems, and demonstrates that metrisableChoquet simplices arise as trace spaces of dynamical QMCS. Collectively, these results provide a bridge between noncommutative geometry, classification theory, and dynamical systems, enabling precise control over how quantum metric structures vary under deformations and embeddings. The framework is illustrated through examples such as tracial rotation algebras and projectionless models, highlighting the interplay between trace geometry, -theory, and continuous variation of dynamics and metrics.

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 -algebra whose trace space is equipped with a metric inducing the -topology, such that tracially Lipschitz elements are dense in . This added structure is designed for measuring distances in and around the category of stably finite classifiable -algebras, and in particular for witnessing metric and statistical properties of the space of approximate unitary equivalence classes of unital embeddings of into a stably finite classifiable -algebra . As for examples, we recall the construction of classifiable -algebraic quantum metric Bauer simplices that function as noncommutative spaces of observables of compact connected metric spaces . We also explain how to build non-Bauer examples by forming 'tracial quantum crossed products' associated with topological dynamical systems on , 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-Hausdorff 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.
Paper Structure (16 sections, 22 theorems, 92 equations)

This paper contains 16 sections, 22 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Quantum spaces
  3. Compact quantum metric spaces
  4. Quantum metric Choquet simplices
  5. Quantum metric Bauer simplices
  6. Metric analysis
  7. Unitary distances
  8. Birkhoff convergence rates
  9. Quantum distances
  10. Quantum systems
  11. $\mathrm{C}^*$-algebras arising as tracial quantum crossed products
  12. Inheritance of QMCS structure
  13. Random systems
  14. Quantum fields
  15. Metric deformation
  16. ...and 1 more sections

Key Result

Theorem 1

For every compact metric space $(X,\rho)$, there is a $\mathrm{C}^*$-algebraic quantum metric Bauer simplex $(\mathcal{A}_X,L_\rho)$ such that $\mathcal{A}_X$ is unital and classifiable with extreme tracial boundary $\partial_e(T(\mathcal{A}_X))$ homeomorphic to $X$, and such that the seminorm $L_\r

Theorems & Definitions (65)

  • Theorem : Jacelon:2022wr
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 55 more