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 \(C^{*}\) $C_{}^{\ast }$ -dynamical system, by using an analogue of the Fourier series representation of a function in the (commutative) torus \(C^\infty ({\mathbb {T}}^n)\) $C^{\infty }({\mathbb{T}}^n)$ , as Rieffel deformations of the standard pseudodifferential operators on \(C^\infty ({\mathbb {T}}^n)\) $C^{\infty }({\mathbb{T}}^n)$ , and in association to certain spectral triples.