  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.

Operator Algebras