Characterisations for coarse embeddability via quasi-locality and Schur multipliers

We establish three new characterisations of coarse embeddability into Hilbert space for discrete metric spaces of bounded geometry, in terms of escaping quasi-local couplings, Schur multipliers, and operator norm state localisation. As applications, we prove that if a sparse metric space of bounded geometry is coarsely embeddable into Hilbert space, then every ghost projection in the quasi-local algebra is compact, while we also provide a counterexample to the converse implication. Finally, we characterise coarse embeddability into Hilbert space via the relative Haagerup property for the quasi-local algebra, and investigate its relation to coarse embeddability into ${\mathrm{\ell }}^p$ for $p>2$.