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

TL;DR

This paper investigates whether the direct sum of matrix algebras can embed into certain operator algebras associated with metric spaces, providing definitive answers to a recent open problem.

Contribution

It resolves a recent open problem by showing the direct sum of matrix algebras does not embed into the uniform Roe algebra but does embed into the quasi-local algebra.

Findings

The $igoplus_ ext{ell_ ext{infty}}$ matrix algebras do not embed into the uniform Roe algebra.

The same direct sum embeds into the quasi-local algebra.

The inclusion of the uniform Roe algebra into the quasi-local algebra can be proper.

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. Hence the inclusion of the uniform Roe algebra into the quasi-local algebra can be proper.