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A note on the quasi-local algebra of expander graphs

Bruno M. Braga, Ján Špakula, Alessandro Vignati

Abstract

We show that the quasi-local algebra of a coarse disjoint union of expander graphs does not contain a Cartan subalgebra isomorphic to $\ell_\infty$. N. Ozawa has recently shown that these algebras are distinct from the uniform Roe algebras of expander graphs, and our result describes a further difference.

A note on the quasi-local algebra of expander graphs

Abstract

We show that the quasi-local algebra of a coarse disjoint union of expander graphs does not contain a Cartan subalgebra isomorphic to . N. Ozawa has recently shown that these algebras are distinct from the uniform Roe algebras of expander graphs, and our result describes a further difference.

Paper Structure

This paper contains 3 sections, 8 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Expander graphs and Cartan subalgebras
  3. Extension of isomorphisms between uniform Roe algebras

Key Result

Theorem 1.1

Let $X=\bigsqcup_{n\in\mathbb{N}}X_n$ be the coarse disjoint union of expander graphs. Then $\mathrm{C}^*_{\!\mathit{ql}}(X)$ does not have a Cartan subalgebra isomorphic to $\ell_\infty$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 2.1: Concentration of measure phenomenon
  • Lemma 2.2
  • proof
  • Claim 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • ...and 15 more