  The rigidity problem for uniform Roe algebras was recently positively solved.
Before its solution was found, there were positive solutions under the
assumption of certain technical geometric conditions. In this paper, we
introduce weaker versions of the operator norm localization property (ONL)
which turn out to characterize those technical geometric conditions. We use
this to obtain new rigidity results for nonmetrizable coarse spaces. As an
application, we provide a novel partial answer to a question of White and
Willett about Cartan subalgebras of uniform Roe algebras. We also study
embeddings between uniform Roe algebras.
