Pseudo-differential Operators and Regularity of Spectral Triples

TL;DR

This paper introduces a new algebraic framework for pseudo-differential operators and establishes a criterion for the regularity of spectral triples, including their stability under products.

Contribution

It defines an algebra of generalized pseudo-differential operators and proves the equivalence with spectral triple regularity, also showing product stability.

Findings

Spectral triples are regular iff they admit an algebra of generalized pseudo-differential operators.

The product of regular spectral triples remains regular.

Provides a self-contained proof of product regularity.

Abstract

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.