Spectral triple

Summary

A spectral triple $(A, H, D)$ consists of an algebra $A$ of operators on a Hilbert space $H$ and an unbounded self-adjoint operator $D$ on $H$. The operator $D$ has compact resolvent, and its commutators with elements of $A$ are bounded. It is the noncommutative version of a compact Riemannian spin manifold together with its Dirac operator. From a spectral triple one recovers a distance, a dimension and an integral, so it encodes metric geometry in operator language.