Embeddings of matrix algebras into the universal Roe algebra

Abstract

V. Manuilov introduced the universal Roe algebra, namely the uniform Roe algebra associated with the largest uniformly locally finite coarse structure $\mathrm{Univ}$ on $\mathbb{N}$, together with several variants. We study these algebras through embeddability properties of $\mathrm{C}^*$-algebras. Among other results, we show that the universal Roe algebra admits an embedding from $\prod_n\mathbb{M}_n$, answering a question by Farah. Thus, Ozawa's non-embeddability theorem for $\prod_n\mathbb{M}_n$ in uniform Roe algebras of uniformly locally finite metric spaces does not extend to arbitrary uniformly locally finite coarse spaces. Nevertheless, embeddability properties distinguish the variants introduced by Manuilov and the quasi-local algebra of $\mathrm{Univ}$, apart from the multiplier algebra $M(\mathbb{I})$. As a by-product, we also obtain an operator-norm analogue of Birkhoff's Problem 111: the norm-closed convex hull of permutation operators is exactly the set of doubly substochastic operators. Moreover, we show that every separable $\mathrm{C}^*$-algebra embeds into the quasi-local algebra of $\mathrm{Univ}$. Although we do not know whether the analogous statement holds for the universal Roe algebra, we prove that every separable quasidiagonal $\mathrm{C}^*$-algebra embeds into it, and hence into the reduced $\mathrm{C}^*$-algebra of some étale locally compact second countable groupoid.