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Bijective rigidity of uniform Roe algebras and injectivity of the comparison map

Kostyantyn Krutoy

TL;DR

The paper establishes a tight link between bijective rigidity of uniform Roe algebras and the injectivity of the $0$th HK comparison map $oldsymbol{eta}_0$ for coarse groupoids. It proves that, within an admissible class of uniformly locally finite spaces, injectivity of $oldsymbol{eta}_0$ for all spaces is equivalent to bijective rigidity, and shows unconditional injectivity (and split-injectivity under coarse connectedness) of $oldsymbol{eta}_0$ in the coarse-groupoid setting. A key technical tool is the construction of uniform covering isometries, which yields a natural transformation from uniformly finite homology to $K_0(C^*_u(-))$ and ensures that isomorphisms of uniform Roe algebras induce the same $K$-theory as the associated coarse maps. These results lead to corollaries for property A spaces, coarse groupoids, and finitely generated groups, including rational split-injectivity of $oldsymbol{eta}_0$ and potential extensions to broader classes of Roe-type algebras and ample groupoids.

Abstract

We show that, for uniformly locally finite metric spaces $X$ and $Y$ with isomorphic uniform Roe algebras $C^*_u(X)$ and $C^*_u(Y)$, the existence of a bijective coarse equivalence $f \colon X \to Y$ is equivalent to the injectivity of the $0$th comparison map appearing in the HK conjecture for coarse groupoids. We further prove that the $0$th comparison map is injective unconditionally. Moreover, if the underlying space is coarsely connected, this map is in fact split-injective.

Bijective rigidity of uniform Roe algebras and injectivity of the comparison map

TL;DR

The paper establishes a tight link between bijective rigidity of uniform Roe algebras and the injectivity of the th HK comparison map for coarse groupoids. It proves that, within an admissible class of uniformly locally finite spaces, injectivity of for all spaces is equivalent to bijective rigidity, and shows unconditional injectivity (and split-injectivity under coarse connectedness) of in the coarse-groupoid setting. A key technical tool is the construction of uniform covering isometries, which yields a natural transformation from uniformly finite homology to and ensures that isomorphisms of uniform Roe algebras induce the same -theory as the associated coarse maps. These results lead to corollaries for property A spaces, coarse groupoids, and finitely generated groups, including rational split-injectivity of and potential extensions to broader classes of Roe-type algebras and ample groupoids.

Abstract

We show that, for uniformly locally finite metric spaces and with isomorphic uniform Roe algebras and , the existence of a bijective coarse equivalence is equivalent to the injectivity of the th comparison map appearing in the HK conjecture for coarse groupoids. We further prove that the th comparison map is injective unconditionally. Moreover, if the underlying space is coarsely connected, this map is in fact split-injective.
Paper Structure (7 sections, 20 theorems, 96 equations)

This paper contains 7 sections, 20 theorems, 96 equations.

Table of Contents

  1. Preliminaries
  2. Coarse geometry
  3. Uniform Roe algebras and rigidity
  4. Uniformly finite homology
  5. Uniform covering isometries
  6. Rigidity of uniform Roe algebras versus injectivity of $\alpha_0$
  7. Consequences

Key Result

Theorem 1

Let $(X,d)$ and $(Y,\partial)$ be uniformly locally finite metric spaces. Then the $C^*$-algebras $C^*_u(X)$ and $C^*_u(Y)$ are Morita equivalent if and only if $(X,d)$ and $(Y,\partial)$ are coarsely equivalent.

Theorems & Definitions (45)

  • Conjecture 1
  • Theorem 1: baudier2022uniform
  • Theorem 2: whyte99amenablity
  • Conjecture 2: HK conjecture
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Definition 1.1
  • Definition 1.2
  • ...and 35 more