A rigidity framework for Roe-like algebras
Diego Martínez, Federico Vigolo
TL;DR
This work develops a unified, self-contained framework to establish stable *-rigidity for Roe-like Roe algebras of countably generated coarse spaces, proving that stable isomorphisms imply coarse equivalences between the underlying spaces. Central to the method are three pillars: spatial implementation of isomorphisms by a covering unitary, a broad uniformization principle ensuring approximate/ quasi-control of maps between Roe-like algebras, and a construction of coarse-approximation relations that translate operator-theoretic data into coarse maps. A refined rigidity theory is then developed via coarse supports, establishing a functorial correspondence between isomorphisms and coarse equivalences, and yielding powerful corollaries about outer automorphism groups and multiplier relationships among Roe algebras. The conclusions unify existing rigidity results across Roe, cp, ql, and uniform Roe algebras, and extend rigidity to groups, semigroups, and inverse semigroups, with significant implications for the interplay between coarse geometry and operator algebras. The framework also highlights the role of quasi-proper operators in bridging local-compactness properties with rigidity phenomena, enabling a robust categorical perspective on the coarse-geometric content of Roe-like algebras.
Abstract
In this memoir we develop a framework to study rigidity problems for Roe-like C*-algebras of countably generated coarse spaces. The main goal is to give a complete and self-contained solution to the problem of C*-rigidity for proper (extended) metric spaces. Namely, we show that (stable) isomorphisms among Roe algebras always give rise to coarse equivalences. The material is organized as to provide a unified proof of C*-rigidity for Roe algebras, algebras of operators of controlled propagation, and algebras of quasi-local operators. We also prove a more refined C*-rigidity statement which has several additional applications. For instance, we can put the correspondence between coarse geometry and operator algebras in a categorical framework, and we prove that the outer automorphism groups of these C*-algebras are all isomorphic to the group of coarse equivalences of the coarse space.