Pseudodifferential Operators on Noncommutative Tori: a Survey

This article presents a survey of recent developments on pseudodifferential operators on noncommutative tori. We describe currently available constructions of those operators: by means of a $C^{\ast }$–dynamical system, by using an analogue of the Fourier series representation of a function in the (commutative) torus $C^{\mathrm{\infty }}({\mathbb{T}}^n)$, as Rieffel deformations of the standard pseudodifferential operators on $C^{\mathrm{\infty }}({\mathbb{T}}^n)$, and in association to certain spectral triples.