We develop an analogoue of coarse geometry for noncommutative spaces in
terms of unitizations of the given C* -algebra. Examples for our theory come
from Rieffel deformation of compactifications under strongly continuous actions
of R^d. A special case of this is the coarse structure on the Moyal plane, seen
as a Rieffel deformation of the classical plane. The motivating question for
this project has been to investigate a possible coarse equivalence between the
classical plane and the Moyal plane, which seems plausible in physics. We define
a noncommutative analogue of coarse maps. Our definition ensures that the
classical and the Moyal plane with their standard coarse structures are coarsely
equivalent. A more general result holds for Rieffel deformations of arbitrary
actions of R^d by translations.
