A rigidity framework for Roe-like algebras

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-like algebras always give rise to coarse equivalences. To do so, we construct a framework to study rigidity problems for Roe-like 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.