Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Noncommutative coarse metric geometry

Ayoub Hafid

Abstract

Motivated by coarse geometry and the classical role of Roe algebras as large-scale invariants of proper metric spaces, we show that proper quantum metric spaces as introduced by Latrémolière are noncommutative coarse spaces. This further allows us to develop a bridge between Latrémolière's framework and the W*-metric approach to quantum metric spaces. Furthermore, we construct Roe algebras for locally compact quantum metric spaces and verify that they recover the classical Roe algebras in the commutative case. We furthermore apply this framework to some examples of locally compact quantum metric spaces and show that it leads to the natural conclusion. Finally we use this framework to introduce notions of higher index theory for locally compact quantum metric spaces.

Noncommutative coarse metric geometry

Abstract

Motivated by coarse geometry and the classical role of Roe algebras as large-scale invariants of proper metric spaces, we show that proper quantum metric spaces as introduced by Latrémolière are noncommutative coarse spaces. This further allows us to develop a bridge between Latrémolière's framework and the W*-metric approach to quantum metric spaces. Furthermore, we construct Roe algebras for locally compact quantum metric spaces and verify that they recover the classical Roe algebras in the commutative case. We furthermore apply this framework to some examples of locally compact quantum metric spaces and show that it leads to the natural conclusion. Finally we use this framework to introduce notions of higher index theory for locally compact quantum metric spaces.
Paper Structure (41 sections, 40 theorems, 147 equations)

This paper contains 41 sections, 40 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Quantum metric spaces
  3. Coarse geometry
  4. Seminorms on Roe algebras: Relative commutants
  5. Summary of results
  6. Preliminaries
  7. Locally Compact Quantum Metric Spaces
  8. Noncommutative coarse geometry
  9. W*-quantum metric spaces
  10. Roe algebras: Propagation and Commutation seminorm quantum metric spaces
  11. Spectral Roe algebras
  12. Definition and first properties
  13. Relative commutant Roe algebra
  14. Finite asymptotic dimension : equality of the algebras
  15. Assouad-Nagata dimension : controlling propagation from commutation semi-norm
  16. ...and 26 more sections

Key Result

Theorem 2.2

If $\mathcal{S}$ is a set of Borel probability measures on a locally compact metric space $(X,d)$ such that for some $x_0\in X$: then the Monge-Kantorovich metric $\mathrm{mk}_{L}$ metrizes the weak* topology on $\mathcal{S}$.

Theorems & Definitions (122)

  • Definition 2.1: Lipschitz pair
  • Theorem 2.2: Dobrushin dobrushinPrescribingSystemRandom1970a
  • Definition 2.3: quantum compact metric spaces Rieffel1999
  • Definition 2.4: Lipschitz triple
  • Definition 2.5: Local states latremoliereQuantumLocallyCompact
  • Definition 2.6: Tame subsets of the state space latremoliereQuantumLocallyCompact
  • Definition 2.7: locally compact quantum metric spaces latremoliereQuantumLocallyCompact
  • Remark 2.8
  • Theorem 2.9: See latremoliereQuantumLocallyCompact
  • Remark 2.10
  • ...and 112 more