  Given a coarse space $(X,\mathcal{E})$, one can define a
$\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of
$(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willett that if
the uniform Roe algebras of two uniformly locally finite metric spaces with
property A are isomorphic, then the metric spaces are coarsely equivalent to
each other. In this paper, we look at the problem of generalizing this result
for general coarse spaces and on weakening the hypothesis of the spaces having
property A.
