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Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence

Jacopo Bassi, Roberto Conti, Carla Farsi, Frederic Latremoliere

Abstract

In this paper we study the groups of isometries and the set of bi-Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latremoliere. In particular we prove that these groups and sets are compact in the automorphism group of the spectral triple C*-algebra with respect to the Monge-Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov-Hausdorff propinquity -- a noncommutative analogue of the Gromov-Hausdorff distance -- when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence

Abstract

In this paper we study the groups of isometries and the set of bi-Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latremoliere. In particular we prove that these groups and sets are compact in the automorphism group of the spectral triple C*-algebra with respect to the Monge-Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov-Hausdorff propinquity -- a noncommutative analogue of the Gromov-Hausdorff distance -- when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.
Paper Structure (12 sections, 25 theorems, 131 equations)

This paper contains 12 sections, 25 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. The Isometry Groups of a Metric Spectral Triple
  3. The Isometry Group of a Quantum Compact Metric Space
  4. The Isometry Group of a Spectral Triple
  5. The Propinquity and Isometry Groups
  6. The Covariant Propinquity
  7. Covariant Bridge Builders
  8. Applications
  9. The Identity as a Bridge Builder
  10. Almost Inners
  11. AF Algebras with Faithful Traces
  12. Dual Actions and Inductive Limits

Key Result

Theorem 1

(Theorem compact-Iso-thm, Corollary cor:ISO-compact, and Theorem compact-iso-group-thm) Let $({\mathfrak{A}},{\mathsf{L}})$ be a quantum compact metric space. Let ${\mathrm{Aut}\left({{\mathfrak{A}}}\right)}$ be the group of $\ast$-automorphisms of ${\mathfrak{A}}$, whose topology of pointwise conve The following subsets and subgroups of $({\mathrm{Aut}\left({{\mathfrak{A}}}\right)}, \mathrm{mkD}_

Theorems & Definitions (58)

  • Theorem
  • Theorem : Theorems \ref{['covariant-bridge-builder-thm']} and \ref{['covariant-bridge-builder-converse-thm']}
  • Theorem
  • Definition 2.2: Connes89Rieffel98aRieffel99Latremoliere13Latremoliere15
  • Definition 2.5: Rieffel99Rieffel00Latremoliere16b
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8: Latremoliere16b
  • Theorem 2.9: Latremoliere16b
  • Lemma 2.10
  • ...and 48 more