# The Theory of Pseudo-differential Operators on the Noncommutative n-Torus

## TL;DR

This paper rigorously develops the pseudo-differential calculus on noncommutative n-tori, filling in technical details, defining Sobolev spaces, and extending results to modules, thereby advancing the mathematical foundation of noncommutative geometry.

## Contribution

It provides detailed proofs and extends the pseudo-differential calculus to noncommutative tori and their modules, strengthening the analytical framework in noncommutative geometry.

## Key findings

- Reproved formulas for symbols of adjoint and product of pseudo-differential operators.
- Defined Sobolev spaces on noncommutative tori and proved related lemmas.
- Extended results to finitely generated projective modules over noncommutative tori.

## Abstract

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on $\mathbb R^n$, one may derive a similar pseudo-differential calculus on noncommutative $n$ tori $\mathbb T_{\theta}^n$, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for $n=2,4$, as shown in the groundbreaking paper in which the Gauss--Bonnet theorem on $\mathbb T_{\theta}^2$ is proved and later papers. Certain details of the proofs in the original derivation of the calculus were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we define the corresponding analog of Sobolev spaces for which we prove the Sobolev and Rellich lemmas. We then extend these results to finitely generated projective right modules over the noncommutative $n$ torus.