Connes' distance function on one-dimensional lattices

TL;DR

This paper demonstrates how Connes' distance function can be used to recover the standard geometry of a one-dimensional equidistant lattice through a specific operator with geometric significance.

Contribution

It introduces an operator that, when applied with Connes' distance function, reproduces the geometry of a linear lattice, linking noncommutative geometry to classical lattice structures.

Findings

Connes' distance function can recover lattice geometry.

A specific operator encodes geometric information.

The approach bridges noncommutative and classical geometry.

Abstract

We show that there is an operator with a simple geometric significance which yields the ordinary geometry of a linear equidistant lattice via Connes' distance function.