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Spectral Triples on noncommutative solenoids from the standard spectral triples on quantum tori

Carla Farsi, Frederic Latremoliere, Judith Packer

Abstract

We address the natural question: as noncommutative solenoids are inductive limits of quantum tori, do the standard spectral triples on quantum tori converge to some spectral triple on noncommutative solenoid for the spectral propinquity? We answer this question by showing that, using appropriate bounded perturbation of the spectral triples on quantum tori, such a spectral triple on noncommutative solenoid can be constructed.

Spectral Triples on noncommutative solenoids from the standard spectral triples on quantum tori

Abstract

We address the natural question: as noncommutative solenoids are inductive limits of quantum tori, do the standard spectral triples on quantum tori converge to some spectral triple on noncommutative solenoid for the spectral propinquity? We answer this question by showing that, using appropriate bounded perturbation of the spectral triples on quantum tori, such a spectral triple on noncommutative solenoid can be constructed.
Paper Structure (3 sections, 10 theorems, 57 equations)

This paper contains 3 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. A Metric Spectral Triple on Noncommutative Solenoids
  3. $K$-homology for noncommutative $2$-solenoids and metric spectral triples

Key Result

Lemma 2.3

$(C^\ast(G_n,\sigma),\ell^2(G_n)\otimes E,{\slashed{D}}_n)$ is a spectral triple for all $n\in {\mathds{N}}\cup\{\infty\}$.

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 10 more