Noncommutative coarse geometry
  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. 
Tathagata Banerjee and Ralf Meyer
math.OA