Reconstruction of manifolds in noncommutative geometry

TL;DR

This paper demonstrates that under certain conditions, the algebra in a commutative spectral triple corresponds exactly to the algebra of smooth functions on a compact spin manifold, linking noncommutative geometry to classical geometry.

Contribution

It establishes that the algebra of a spectral triple satisfying specific conditions is isomorphic to the algebra of smooth functions on a compact spin manifold, strengthening the connection between noncommutative and classical geometry.

Findings

The algebra A corresponds to smooth functions on a compact spin manifold.

Additional conditions ensure the algebra's classical geometric interpretation.

The result bridges noncommutative spectral triples with classical manifold structures.

Abstract

We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.