A rigidity framework for Roe-like algebras

Diego Martínez; Federico Vigolo

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arXiv · 2403.13624

A rigidity framework for Roe-like algebras

Diego Martínez·Federico Vigolo

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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.

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Diego Martínez, Federico Vigolo. 2024-03-20. A rigidity framework for Roe-like algebras. https://arxiv.org/abs/2403.13624

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