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Gromov-Hausdorff convergence of metric spaces of UCP maps

Tirthankar Bhattacharyya, Ritul Duhan, Chandan Pradhan

TL;DR

The paper addresses how to recover geometric information from truncated operator-system data by analyzing sets of unital completely positive maps. It extends van Suijlekom’s BW-condition framework to obtain Gromov–Hausdorff convergence of CP-map spaces via $C^1$-approximate complete order isomorphisms between operator system spectral triples, showing convergence of truncations like $C(S^1)_{(n)}$ and $\mathcal{T}_n$ to $C(S^1)$. Concrete applications include Fejér–Riesz operator systems, Toeplitz operator systems, and truncations of the $d$-torus, with spherical and polyhedral schemes all yielding GH convergence of $\mathcal{UCP}_{\mathcal{K}}(E_n)$ to $\mathcal{UCP}_{\mathcal{K}}(E)$. The results provide a principled way to extract geometric information from partial spectral data in noncommutative geometry and related computational contexts, highlighting the robustness of the BW-topology-based approach for metric convergence.

Abstract

It is shown that van Suijlekom's technique of imposing a set of conditions on operator system spectral triples ensures Gromov-Hausdorff convergence of sequences of sets of unital completely positive maps (equipped with the BW-topology which is metrizable). This implies that even when only a part of the spectrum of the Dirac operator is available together with a certain truncation of the $C^*$-algebra, information about the geometry can be extracted.

Gromov-Hausdorff convergence of metric spaces of UCP maps

TL;DR

The paper addresses how to recover geometric information from truncated operator-system data by analyzing sets of unital completely positive maps. It extends van Suijlekom’s BW-condition framework to obtain Gromov–Hausdorff convergence of CP-map spaces via -approximate complete order isomorphisms between operator system spectral triples, showing convergence of truncations like and to . Concrete applications include Fejér–Riesz operator systems, Toeplitz operator systems, and truncations of the -torus, with spherical and polyhedral schemes all yielding GH convergence of to . The results provide a principled way to extract geometric information from partial spectral data in noncommutative geometry and related computational contexts, highlighting the robustness of the BW-topology-based approach for metric convergence.

Abstract

It is shown that van Suijlekom's technique of imposing a set of conditions on operator system spectral triples ensures Gromov-Hausdorff convergence of sequences of sets of unital completely positive maps (equipped with the BW-topology which is metrizable). This implies that even when only a part of the spectrum of the Dirac operator is available together with a certain truncation of the -algebra, information about the geometry can be extracted.

Paper Structure

This paper contains 12 sections, 11 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Sets of CP maps as metric spaces
  3. The BW-topology
  4. A metric for the BW-topology when $E = C(\Omega)$.
  5. A metric for the BW-topology when $E$ is a finite-dimensional operator system
  6. The Gromov--Hausdorff convergence
  7. Applications
  8. The Fejér--Riesz operator systems
  9. The Toeplitz operator systems
  10. The $d$-dimensional torus
  11. Spherical truncation
  12. Polyhedral truncation

Key Result

Lemma 2.2

The function $d$ given by def:metric forms a metric on $\mathcal{UCP}_{\mathcal{K}}(C(\Omega))$.

Theorems & Definitions (23)

  • Definition 2.1: BW-topology
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 13 more