Noncommutative Lattices and Their Continuum Limits

TL;DR

This paper explores how noncommutative lattices can serve as finite approximations of topological spaces and how the original space and its algebra of continuous functions can be reconstructed from these approximations.

Contribution

It demonstrates a method to recover a topological space and its continuous function algebra from systems of noncommutative lattices and $C^*$-algebras.

Findings

Reconstruction of space $M$ from noncommutative lattices

Recovery of algebra $ ext{C}(M)$ from noncommutative $C^*$-algebras

Framework for approximating topological spaces with noncommutative structures

Abstract

We consider finite approximations of a topological space $M$ by noncommutative lattices of points. These lattices are structure spaces of noncommutative $C^*$-algebras which in turn approximate the algebra $\cc(M)$ of continuous functions on $M$. We show how to recover the space $M$ and the algebra $\cc(M)$ from a projective system of noncommutative lattices and an inductive system of noncommutative $C^*$-algebras, respectively.