Inverse Limits of Spectral Triples

TL;DR

This paper explores the noncommutative generalization of the relationship between Riemannian manifolds and spectral triples, extending classical geometric concepts to noncommutative geometry.

Contribution

It introduces the concept of inverse limits of spectral triples, generalizing the notion of coverings of Riemannian manifolds to the noncommutative setting.

Findings

Established a framework for inverse limits of spectral triples.

Connected noncommutative $C^*$-algebras with generalized geometric structures.

Extended classical covering space theory to noncommutative geometry.

Abstract

Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Similarly a spectral triple is a generalization of a Riemannian manifold. An (infinitely listed) covering of a Riemannian manifold has natural structure of Riemannian manifold. Here we will consider the noncommutative generalization of this result.