Dynamical propagation and Roe algebras of warped spaces

Given a non-singular action $\mathrm{\Gamma }\curvearrowright (X,\mu )$, we define the $\ast$-algebra ${\mathbb{C}}_{\mathrm{f}\mathrm{p}}[\mathrm{\Gamma }\curvearrowright X]$ 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 $L^{\mathrm{\infty }}X{⋊}_{\mathrm{a}\mathrm{l}\mathrm{g}}\mathrm{\Gamma }$ surjects onto ${\mathbb{C}}_{\mathrm{f}\mathrm{p}}[\mathrm{\Gamma }\curvearrowright X]$ 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 ${\mathbb{C}}_{\mathrm{f}\mathrm{p}}[\mathrm{\Gamma }\curvearrowright X]$ 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.