Asymptotic coarsening of proper metric spaces and index problems

We introduce an asymptotic coarse structure on proper metric spaces and study the associated C*-algebras and assembly maps. We establish an asymptotic Lipschitz homotopy invariance theorem for the K-theory of these C*-algebras and the K-homology of the metric space, and show that the assembly map is an isomorphism over an asymptotically scaleable space.