Quantum Locally Compact Metric Spaces

TL;DR

This paper introduces quantum locally compact metric spaces, extending Rieffel's quantum metric spaces to nonunital cases, with examples like the Moyal plane, advancing noncommutative geometry for noncompact quantum spaces.

Contribution

It generalizes the concept of quantum metric spaces to nonunital settings and provides concrete examples, addressing a key open question in noncommutative geometry.

Findings

Defined quantum locally compact metric spaces

Provided examples including the Moyal plane

Connected to classical locally compact metric spaces

Abstract

We introduce the notion of a quantum locally compact metric space, which is the noncommutative analogue of a locally compact metric space, and generalize to the nonunital setting the notion of quantum metric spaces introduced by Rieffel. We then provide several examples of such structures, including the Moyal plane, as well as compact quantum metric spaces and locally compact metric spaces. This paper provides an answer to the question raised in the literature about the proper notion of a quantum metric space in the nonunital setup and offers important insights into noncommutative geometry for non compact quantum spaces.