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Rigidity for geometric ideals in uniform Roe algebras

Baojie Jiang, Jiawen Zhang

Abstract

In this paper, we investigate the rigidity problems for geometric ideals in uniform Roe algebras associated to discrete metric spaces of bounded geometry. These ideals were introduced by Chen and Wang, and can be fully characterised in terms of ideals in the associated coarse structures. Our main result is that if two geometric ideals in uniform Roe algebras are stably isomorphic, then the coarse spaces associated to these ideals are coarsely equivalent. We also discuss the case of ghostly ideals and pose some open questions.

Rigidity for geometric ideals in uniform Roe algebras

Abstract

In this paper, we investigate the rigidity problems for geometric ideals in uniform Roe algebras associated to discrete metric spaces of bounded geometry. These ideals were introduced by Chen and Wang, and can be fully characterised in terms of ideals in the associated coarse structures. Our main result is that if two geometric ideals in uniform Roe algebras are stably isomorphic, then the coarse spaces associated to these ideals are coarsely equivalent. We also discuss the case of ghostly ideals and pose some open questions.
Paper Structure (14 sections, 39 theorems, 78 equations)

This paper contains 14 sections, 39 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard notation
  4. Notions from coarse geometry
  5. Uniform Roe algebras and geometric ideals
  6. Coarse correspondence
  7. The unital case
  8. The general case
  9. Coarse equivalences induce Morita equivalences
  10. Rigidity for geometric ideals
  11. Rigid isomorphisms for geometric ideals
  12. From isomorphisms to rigid isomorphisms
  13. Rigidity for stable geometric ideals
  14. Discussions on ghostly ideals

Key Result

Proposition 4

Two coarse spaces $(X, \mathscr{E}_X)$ and $(Y, \mathscr{E}_Y)$ are coarsely equivalent if and only if there exist coarse families (see Definition defn:coarse family) of maps such that $\{(g_{f_L(L)} \circ f_L(x), x) ~|~ x\in L\} \in \mathscr{E}_X$ and $\{(f_{g_{L'}(L')} \circ g_{L'}(y), y) ~|~ y\in L'\} \in \mathscr{E}_Y$ for any $L \in \mathbf{L}(\mathscr{E}_X)$ and $L' \in \mathbf{L}(\mathscr{

Theorems & Definitions (85)

  • Definition 1: CW04Wan07
  • Definition 3: STY02
  • Proposition 4: Corollary \ref{['cor:coarse equiv general case']}
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: CW04
  • ...and 75 more