Embeddings of matrix algebras into uniform Roe algebras and quasi-local algebras

We answer the recent problem posed by Baudier, Braga, Farah, Vignati, and Willett that asks whether the ${\mathrm{\ell }}_{\mathrm{\infty }}$-direct sum of the matrix algebras embeds into the uniform Roe algebra or the quasi-local algebra of a uniformly locally finite metric space. The answers are no and yes, respectively. Hence the inclusion of the uniform Roe algebra into the quasi-local algebra can be proper.