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Gromov--Hausdorff Convergence of Spectral Truncations for Tori

Malte Leimbach, Walter D. van Suijlekom

Abstract

We consider operator systems associated to spectral truncations of tori. We show that their state spaces, when equipped with the Connes distance function, converge in the Gromov--Hausdorff sense to the space of all Borel probability measures on the torus equipped with the Monge--Kantorovich distance. A crucial role will be played by the relationship between Schur and Fourier multipliers. Along the way, we introduce the spectral Fejér kernel and show that it is a good kernel. This allows to make the estimates sufficient to prove the desired convergence of state spaces. We conclude with some structure analysis of the pertinent operator systems, including the C*-envelope and the propagation number, and with an observation about the dual operator system.

Gromov--Hausdorff Convergence of Spectral Truncations for Tori

Abstract

We consider operator systems associated to spectral truncations of tori. We show that their state spaces, when equipped with the Connes distance function, converge in the Gromov--Hausdorff sense to the space of all Borel probability measures on the torus equipped with the Monge--Kantorovich distance. A crucial role will be played by the relationship between Schur and Fourier multipliers. Along the way, we introduce the spectral Fejér kernel and show that it is a good kernel. This allows to make the estimates sufficient to prove the desired convergence of state spaces. We conclude with some structure analysis of the pertinent operator systems, including the C*-envelope and the propagation number, and with an observation about the dual operator system.
Paper Structure (13 sections, 14 theorems, 68 equations, 1 figure)

This paper contains 13 sections, 14 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Actions and commutators
  4. Good kernels
  5. Fourier and Schur multipliers
  6. Convergence of Spectral Truncations of the $d$-Torus
  7. A candidate for the $\mathrm{C}^1$-approximate order isomorphism
  8. A transference result
  9. The spectral Fejér kernel
  10. Structure Analysis of the Operator System $\mathrm{C}(\mathbb{T}^d)^{(\Lambda)}$
  11. C*-envelope and propagation number
  12. Dual
  13. Some convex geometry

Key Result

Lemma 2.2

If $K_\Lambda$ is a good kernel, then, for all $f \in \mathrm{C}^\infty(\mathbb{T}^d)$, the following holds: where $\gamma_\Lambda \rightarrow 0$ as $\Lambda \rightarrow \infty$.

Figures (1)

  • Figure 1: A plot of the boundaries of the balls $\mathrm{B}_\Lambda$ and the polytopes $K_\Lambda$, for $\Lambda = 0, 1, \sqrt{2}, 2, \sqrt{5}, 2\sqrt{2}, 3$.

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 24 more