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A rigidity framework for Roe-like algebras

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arxiv 2403.13624 v4 pith:6INVH66J submitted 2024-03-20 math.OA math.MG

Diego Martínez , Federico Vigolo This is my paper · ORCID
classification math.OAmath.MG
keywords algebrascoarserigidityframeworkequivalencesgiveoperatorsprove
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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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces

    math.OA 2026-07 accept novelty 8.0 of 10

    Isomorphism of uniform Roe algebras over uniformly locally finite coarse spaces forces bijective coarse equivalence of the underlying spaces.

  2. Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree

    math.KT 2025-04 conditional novelty 8.0 of 10

    For free strongly ergodic actions of property A groups on compact manifolds, the coarse assembly map for the unified warped cone is not surjective in K_1.

  3. C*-rigidity of bounded geometry metric spaces

    math.OA 2025-01 accept novelty 8.0 of 10

    Uniformly locally finite metric spaces with isomorphic Roe algebras are coarsely equivalent, and the outer automorphism group of the Roe algebra is canonically isomorphic to the group of coarse equivalences.

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