Nuclearity noncommutative torus

Question

I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an idea of the proof? I thank you in advance for the help.

Answer 1

If $A$ is nuclear, $G$ is a locally compact amenable group, and $\alpha$ is an action of $G$ on $A$, then $A \rtimes_\alpha G$ is nuclear (see, for instance, Blackadar, Operator Algebras: Theory of $C^\ast$-Algebras and Von Neumann Algebras, Corollary IV.3.5.2). In particular, if $\theta$ is irrational, then $C(\mathbb{T}^2_\theta)$ is nuclear since $C(\mathbb{T}^2_\theta) \cong C(S^1) \rtimes_\theta \mathbb{Z}$, where $\theta$, by slight abuse of notation, denotes rotation by the angle $2\pi\theta$.