Rigidity for geometric ideals in uniform Roe algebras

In this paper, we investigate the rigidity problems for geometric ideals in uniform Roe algebras associated to discrete metric spaces of bounded geometry. These ideals were introduced by Chen and Wang, and can be fully characterised in terms of ideals in the associated coarse structures. Our main result is that if two geometric ideals in uniform Roe algebras are stably isomorphic, then the coarse spaces associated to these ideals are coarsely equivalent. We also discuss the case of ghostly ideals and pose some open questions.