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

Abstract

We answer the recent problem posed by Baudier, Braga, Farah, Vignati, and Willett that asks whether the ${\ell }_{\infty }$-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. This yields the existence of a quasi-local operator that is not approximable by finite propagation operators.