  We study embeddings of uniform Roe algebras which have "large range" in their
codomain and the relation of those with coarse quotients between metric spaces.
Among other results, we show that if $Y$ has property A and there is an
embedding $\Phi:\mathrm{C}^*_u(X)\to \mathrm{C}^*_u(Y)$ with "large range" and
so that $\Phi(\ell_\infty(X))$ is a Cartan subalgebra of $\mathrm{C}^*_u(Y)$,
then there is a bijective coarse quotient $X\to Y$. This shows that the large
scale geometry of $Y$ is, in some sense, controlled by the one of $X$. For
instance, if $X$ has finite asymptotic dimension, so does $Y$.
