Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space $X$. Under weak assumptions, these ${\mathrm{C}}^{\ast }$-algebras contain embedded copies of ${\prod }_k{\mathrm{M}}_{n_k}(\mathbb{C})$ for any \emph{bounded} countable (possibly finite) collection $(n_k{)}_k$ of natural numbers; we aim to show that they cannot contain any other von Neumann algebras. One of our main results shows that $L_{\mathrm{\infty }}[0,1]$ does not embed into any of those algebras, even by a not-necessarily-normal $\ast$-homomorphism. In particular, it follows from the structure theory of von Neumann algebras that any von Neumann algebra which embeds into such algebra must be of the form ${\prod }_k{\mathrm{M}}_{n_k}(\mathbb{C})$ for some countable (possibly finite) collection $(n_k{)}_k$ of natural numbers. Under additional assumptions, we also show that the sequence $(n_k{)}_k$ has to be bounded: in other words, the only embedded von Neumann algebras are the ``obvious’’ ones.