Dynamical propagation and Roe algebras of warped spaces

TL;DR

This paper introduces a new algebraic framework for analyzing non-singular group actions and warped spaces, linking dynamical properties to operator algebra structures and applying these to Roe algebras of warped cones.

Contribution

It defines a canonical algebra of finite dynamical propagation operators and characterizes ergodic properties via algebraic structures, also describing Roe algebras of warped spaces in terms of group actions.

Findings

The algebraic crossed product surjects onto the finite dynamical propagation algebra.

Essential freeness of the action implies a *-isomorphism between the crossed product and the finite propagation algebra.

The structure of the algebra characterizes ergodicity and strong ergodicity.

Abstract

Given a non-singular action $\Gamma \curvearrowright (X,\mu)$, we define the $*$-algebra $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of $\mu$. We prove that the algebraic crossed product $L^{\infty}X \rtimes_{\rm alg} \Gamma$ surjects onto $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ 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 $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ 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.