Dynamical propagation and Roe algebras of warped spaces

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.