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 $\cc(M)$ 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 $C^{\ast }$-algebras, respectively.

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

References in corpus (4)

• Noncommutative Lattices as Finite Approximations and Their Noncommutative Geometries
• Simplicial Quantum Gravity and Random Lattices
• Lattices and Their Continuum Limits
• Finite Approximations to Quantum Physics: Quantum Points and their Bundles

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• A Twisted ${\mathcal{C}}^{\star }$-algebra formulation of Quantum Cosmology with application to the Bianchi I model