  We give a complete solution to the problem of C*-rigidity for (extended)
locally compact metric spaces. Namely, we show that (stable) isomorphisms among
Roe algebras always give rise to coarse equivalences. To do so, we develop a
framework to study rigidity problems for Roe-like C*-algebras of countably
generated coarse spaces. This allows us to provide a unified proof of
C*-rigidity for Roe algebras, algebras of operators of controlled propagation,
and algebras of quasi-local operators. As further applications, we prove that
the outer automorphism groups of all of these algebras are isomorphic to the
group of coarse equivalences of the starting coarse space.
