Noncommutative coarse geometry

TL;DR

This paper develops a framework for noncommutative coarse geometry using C*-algebra compactifications, establishing coarse equivalences under Rieffel deformations and actions, and defining noncommutative coarse maps via completely positive maps.

Contribution

It introduces a novel approach to noncommutative coarse geometry, extending classical concepts to C*-algebras and group actions, with new results on coarse equivalences and map liftings.

Findings

Noncommutative coarse structures can be transferred via Rieffel deformations.

Cocompact actions induce coarse structures equivalent to standard group coarse structures.

Noncommutative coarse maps can be constructed from completely positive maps, with lifting theorems for boundary maps.

Abstract

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.