Dynamical propagation and Roe algebras of warped space

Given a non-singular action Γ↷(X,μ), we define the ∗-algebra Cfp[Γ↷X] of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and depends only on the class of measures of μ. We prove that the algebraic crossed product L∞X⋊algΓ surjects onto Cfp[Γ↷X] 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 Cfp[Γ↷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.