Pseudo-differential Operators and Regularity of Spectral Triples

We introduce a notion of an algebra of generalized pseudo-differential operators and prove that a spectral triple is regular if and only if it admits an algebra of generalized pseudo-differential operators. We also provide a self-contained proof of the fact that the product of regular spectral triples is regular.

💡 Research Summary

The paper addresses a central structural issue in non‑commutative geometry: the regularity of spectral triples. Regularity, originally defined analytically as the requirement that the algebra (\mathcal{A}) and its commutators with the Dirac operator (D) lie in the intersection of the domains of all iterated derivations (\delta^{k}) (with (\delta(T)=