Noncommutative Lattices and Their Continuum Limits

Abstract

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

Comments: 22 pages, 8 Figures included in the LaTeX Source New version, minor modifications (typos corrected) and a correction in the list of authors