On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces

Given a coarse space $(X,\mathcal{E})$, one can define a ${\mathrm{C}}^{\ast }$-algebra ${\mathrm{C}}_u^{\ast }(X)$ 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.