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Dynamical propagation and Roe algebras of warped spaces

Tim de Laat, Federico Vigolo, Jeroen Winkel

Abstract

Given a non-singular action $Γ\curvearrowright (X,μ)$, we define the $*$-algebra $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of $μ$. We prove that the algebraic crossed product $L^{\infty}X \rtimes_{\rm alg} Γ$ surjects onto $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ and that this surjection is a $\ast$-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones.

Dynamical propagation and Roe algebras of warped spaces

Abstract

Given a non-singular action , we define the -algebra of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of . We prove that the algebraic crossed product surjects onto and that this surjection is a -isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones.
Paper Structure (11 sections, 30 theorems, 78 equations)

This paper contains 11 sections, 30 theorems, 78 equations.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES: ACTIONS AND ALGEBRAS
  3. Notation and conventions
  4. Actions on measure spaces
  5. Crossed products
  6. PROOF OF THEOREM \ref{['thmA:finite propagation as crossed product']}
  7. PRELIMINARIES: ROE ALGEBRAS AND WARPED SPACES
  8. Roe algebras
  9. Warped spaces
  10. ROE ALGEBRAS OF (WARPED) METRIC SPACES
  11. ROE ALGEBRAS OF WARPED CONES

Key Result

Theorem 1

The $\ast$/̄homomorphism $\varPhi\colon L^\infty X\rtimes_{\rm alg}\Gamma\to\mathbb{C}_{\rm fp}[\Gamma\curvearrowright X]$ is surjective. If the action $\alpha \colon \Gamma \curvearrowright X$ is essentially free, then $\varPhi$ is a $\ast$/̄isomorphism.

Theorems & Definitions (69)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Remark 1.3
  • Remark 1.4
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • ...and 59 more