A note on the quasi-local algebra of expander graphs

We show that the quasi-local algebra of a coarse disjoint union of expander graphs does not contain a Cartan subalgebra isomorphic to ${\mathrm{\ell }}_{\mathrm{\infty }}$. 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.