  We use compactifications of C*-algebras to introduce noncommutative coarse
geometry. We transfer a noncommutative coarse structure on a C*-algebra with an
action of a locally compact Abelian group by translations to Rieffel
deformations and prove that the resulting noncommutative coarse spaces are
coarsely equivalent. We construct a noncommutative coarse structure from a
cocompact continuously square-integrable action of a group and show that this
is coarsely equivalent to the standard coarse structure on the group in
question. We define noncommutative coarse maps through certain completely
positive maps that induce *-homomorphisms on the boundaries of the
compactifications. We lift *-homomorphisms between separable, nuclear
boundaries to noncommutative coarse maps and prove an analogous lifting theorem
for maps between the metrisable boundaries of ordinary locally compact spaces.
