A quantization of coarse spaces and uniform Roe algebras

TL;DR

This paper introduces a quantum version of coarse spaces and uniform Roe algebras, expanding the classical framework with new examples and a developed theory for maps between quantum coarse spaces.

Contribution

It develops a quantization framework for coarse spaces and uniform Roe algebras, including new classes like support expansion C*-algebras and a theory for quantum maps.

Findings

Quantum coarse spaces generalize classical structures.

Introduction of support expansion C*-algebras.

Development of a theory for maps between quantum coarse spaces.

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 l_infty(X) subset B(l_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*-algebras. We also work out the basic theory for maps between quantum coarse spaces and their consequences for quantum uniform Roe algebras.