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C*-rigidity of bounded geometry metric spaces

Diego Martínez, Federico Vigolo

TL;DR

The paper solves the $*$-rigidity problem for bounded geometry spaces by showing that isomorphic Roe algebras $C^*_{ m Roe}(X)$ and $C^*_{ m Roe}(Y)$ imply coarse equivalence of $X$ and $Y$, with the same for uniform Roe and band-dominated algebras. The proof hinges on an unconditional concentration inequality derived from the $cotype$-2 property of Hilbert spaces and the fact that isomorphisms between Roe-type algebras are spatially implemented by unitaries; this yields a coarse map that captures the large-scale geometry. A key refinement identifies $ ext{CE}(X)$ with $ ext{Out}(C^*_{ m Roe}(X))$, extends to $ ext{Out}(C^*_{ m cp}(X))$, and shows that automorphisms are canonically realized by coarse equivalences up to closeness, establishing a strong functorial bridge between coarse geometry and operator algebras. These results enhance the understanding of how large-scale geometric data is encoded in Roe-like algebras and have potential implications for index theory, geometric group theory, and mathematical physics. All conclusions are stated for spaces of bounded geometry, with methods potentially adaptable to arbitrary proper spaces.

Abstract

We prove that uniformly locally finite metric spaces with isomorphic Roe algebras must be coarsely equivalent. As an application, we also prove that the outer automorphism group of the Roe algebra of a metric space of bounded geometry is canonically isomorphic to the group of coarse equivalences of the space up to closeness.

C*-rigidity of bounded geometry metric spaces

TL;DR

The paper solves the -rigidity problem for bounded geometry spaces by showing that isomorphic Roe algebras and imply coarse equivalence of and , with the same for uniform Roe and band-dominated algebras. The proof hinges on an unconditional concentration inequality derived from the -2 property of Hilbert spaces and the fact that isomorphisms between Roe-type algebras are spatially implemented by unitaries; this yields a coarse map that captures the large-scale geometry. A key refinement identifies with , extends to , and shows that automorphisms are canonically realized by coarse equivalences up to closeness, establishing a strong functorial bridge between coarse geometry and operator algebras. These results enhance the understanding of how large-scale geometric data is encoded in Roe-like algebras and have potential implications for index theory, geometric group theory, and mathematical physics. All conclusions are stated for spaces of bounded geometry, with methods potentially adaptable to arbitrary proper spaces.

Abstract

We prove that uniformly locally finite metric spaces with isomorphic Roe algebras must be coarsely equivalent. As an application, we also prove that the outer automorphism group of the Roe algebra of a metric space of bounded geometry is canonically isomorphic to the group of coarse equivalences of the space up to closeness.
Paper Structure (6 sections, 11 theorems, 39 equations)

This paper contains 6 sections, 11 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of C*-rigidity
  4. A concentration inequality
  5. Completing the proof
  6. More refined results

Key Result

Theorem 1

Let $X$ and $Y$ be uniformly locally finite metric spaces. If $C^*_{\rm Roe}{\left(X\right)}\cong C^*_{\rm Roe}{\left(Y\right)}$, then $X$ and $Y$ are coarsely equivalent. Moreover, the same holds if $C^*_{\rm Roe}{\left(\,\textnormal{-}\,\right)}$ is replaced with $C^*_{\rm u}{\left(\,\textnormal{-

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Definition 2.1
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 24 more