Noncommutative coarse metric geometry

TL;DR

This paper establishes a connection between noncommutative coarse geometry and quantum metric spaces, introduces Roe algebras for quantum spaces, and explores higher index theory within this framework.

Contribution

It develops a noncommutative coarse geometric framework for quantum metric spaces, bridging existing approaches and extending Roe algebra constructions to the quantum setting.

Findings

Proper quantum metric spaces are noncommutative coarse spaces.

Roe algebras for quantum metric spaces recover classical Roe algebras in the commutative case.

Framework enables higher index theory for quantum metric spaces.

Abstract

Motivated by coarse geometry and the classical role of Roe algebras as large-scale invariants of proper metric spaces, we show that proper quantum metric spaces as introduced by Latr\'emoli\`ere are noncommutative coarse spaces. This further allows us to develop a bridge between Latr\'emoli\`ere's framework and the W*-metric approach to quantum metric spaces. Furthermore, we construct Roe algebras for locally compact quantum metric spaces and verify that they recover the classical Roe algebras in the commutative case. We furthermore apply this framework to some examples of locally compact quantum metric spaces and show that it leads to the natural conclusion. Finally we use this framework to introduce notions of higher index theory for locally compact quantum metric spaces.