A Remark on Gelfand Duality for Spectral Triples

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable “metric” category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a “quotient” of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.

💡 Research Summary

The paper establishes a precise categorical duality that extends the classical Gelfand correspondence to the realm of spectral triples, thereby linking the geometry of compact Riemannian spin manifolds with the algebraic structure of commutative pre‑C*‑algebras equipped with metric data. The authors begin by recalling the traditional Gelfand–Naimark duality, which identifies compact Hausdorff spaces with unital commutative C*‑algebras, and then motivate the need for a non‑commutative analogue capable of encoding not only topological but also metric information. In the framework of non‑commutative geometry, a spectral triple (𝔄,ℋ,D) consists of a *‑algebra 𝔄 represented on a Hilbert space ℋ together with a self‑adjoint Dirac‑type operator D. When 𝔄 is the algebra C∞(M) of smooth functions on a compact spin manifold M, ℋ = L²(M,S) is the space of square‑integrable spinors, and D is the classical Dirac operator, the triple reproduces the Riemannian metric via Connes’ distance formula d_D(x,y)=sup{|f(x)-f(y)| : ‖