A quantization of coarse spaces and uniform Roe algebras

Bruno M. Braga, Joseph Eisner and David Sherman

Vol. 338 (2025), No. 1, 163–207

DOI: 10.2140/pjm.2025.338.163

Abstract

We propose a quantization of coarse spaces and uniform Roe algebras. The objects are based on the quantum relations introduced by N. Weaver and require the choice of a represented von Neumann algebra. In the case of the diagonal inclusion ${\ell }_{\infty }(X)\subset \mathcal{ℬ}({\ell }_2(X))$, they reduce to the usual constructions. Quantum metric spaces furnish natural examples parallel to the classical setting, but we provide other examples that are not inspired by metric considerations, including the new class of support expansion ${C<!--FUNCTION\; APPLICATION-->}^{\ast }$-algebras. We also work out the basic theory for maps between quantum coarse spaces and their consequences for quantum uniform Roe algebras.

Keywords

uniform Roe algebras, quantum relations

Mathematical Subject Classification

Primary: 46L65, 46L89, 51F30

Milestones

Received: 13 January 2025

Revised: 20 June 2025

Accepted: 20 June 2025

Published: 8 August 2025