  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 $\mathrm{C}^*$-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 $L_\infty[0,1]$ does not embed into any of
those algebras, even by a not-necessarily-normal $*$-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.
