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The rigidity problem for uniform Roe algebras

Alessandro Vignati

Abstract

We solve the rigidity problem for uniform Roe algebras, by showing that two uniformly locally finite metric spaces with isomorphic uniform Roe algebras are bijectively coarsely equivalent.

The rigidity problem for uniform Roe algebras

Abstract

We solve the rigidity problem for uniform Roe algebras, by showing that two uniformly locally finite metric spaces with isomorphic uniform Roe algebras are bijectively coarsely equivalent.
Paper Structure (11 sections, 21 theorems, 69 equations)

This paper contains 11 sections, 21 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. The strategy
  3. Slowly oscillating functions and their images
  4. The main result
  5. Corollaries and concluding remarks
  6. Roe Cartan subalgebras
  7. Automorphisms of uniform Roe algebras
  8. Concluding remarks
  9. A self-contained proof of bijective rigidity
  10. General coarse structures
  11. Embeddings

Key Result

Theorem A

Let $(X,d_X)$ and $(Y,d_Y)$ be uniformly locally finite metric spaces. If the uniform Roe algebras $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_u(Y)$ are isomorphic, then $X$ and $Y$ are bijectively coarsely equivalent.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 33 more