A note on the quasi-local algebra of expander graphs

TL;DR

This paper demonstrates that the quasi-local algebra of a coarse disjoint union of expander graphs lacks a Cartan subalgebra isomorphic to ill__, highlighting a structural difference from uniform Roe algebras.

Contribution

It establishes a new structural property of quasi-local algebras of expander graphs, showing they do not contain certain Cartan subalgebras, extending previous distinctions from uniform Roe algebras.

Findings

Quasi-local algebra of expander graphs lacks ill__ Cartan subalgebra.

Differentiates quasi-local algebra from uniform Roe algebra.

Highlights structural limitations of quasi-local algebra.

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.