  We generalize all known results on rigidity of uniform Roe algebras to the
setting of arbitrary uniformly locally finite coarse spaces. For instance, we
show that isomorphism between uniform Roe algebras of uniformly locally finite
coarse spaces whose uniform Roe algebras contain only compact ghost projections
implies that the base spaces are coarsely equivalent. Moreover, if one of the
spaces has property A, then the base spaces are bijectively coarsely
equivalent. We also provide a characterization for the existence of an
embedding onto hereditary subalgebra in terms of the underlying spaces. As an
application, we partially answer a question of White and Willett about Cartan
subalgebras of uniform Roe algebras.
