In the work of [Connes and Marcolli](http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2006/1.pdf), on page 20, it state that:



> 
> Just as in the classical case of a (commutative) manifold, what ensures that the derivations
> considered are enough to span the whole tangent space is the condition of ellipticity for the
> Laplacian...
> 
> 
> 


The derivations stated are given by (6.15), which are only two elements.


First thing I don't understand is how to prove this. Second thing I don't understand is, if we consider $\delta(U)=V^{\*}-U^{2}Ve^{2\pi i\theta}$, $\delta(V)=0$, then it is also a derivation, and does not belongs to module spanned by $\delta\_{1},\delta\_{2}$. Am I missing something?



Why Der($A_{\theta}$) is spanned by two elements only?
[Bratteli–Elliott–Jorgensen](https://eudml.org/doc/152592) prove a range of classification results for unbounded derivations on a totally irrational noncommutative torus $C(\mathbb{T}^n\_\theta)$, which basically say that any reasonable $\ast$-derivation will be the sum of a $\mathbb{R}$-linear combination of the infinitesimal generators $\delta\_1,\dotsc,\delta\_n$ of the translation action of $\mathbb{T}^n$ and an (approximately) inner derivation.


Here’s a typical special case, which follows from a combination of results in Bratteli–Elliott–Jorgensen. Let $\theta \in \mathbb{R}$ be irrational, let $U$ and $V$ be the usual unitary generators of the continuous noncommutative $2$-torus $C(\mathbb{T}^2\_\theta)$, and let $\mathcal{O}(\mathbb{T}^2\_\theta)$ be the dense $\ast$-subalgebra of Fourier polynomials in $U$, $V$. Then any $\ast$-derivation $\delta : \mathcal{O}(\mathbb{T}^2\_\theta) \to \mathcal{O}(\mathbb{T}^2\_\theta)$ has a unique decomposition $$\delta = c\_1 \delta\_1 + c\_2 \delta\_2 + \tilde{\delta},$$ where $c\_1,c\_2 \in \mathbb{R}$ and $\tilde{\delta}$ is inner, i.e., there exists self-adjoint $H \in \mathcal{O}(\mathbb{T}^2\_\theta)$, such that $\tilde{\delta}(a) = i[H,a]$ for all $a \in \mathcal{O}(\mathbb{T}^2\_\theta)$.


Please note that your would-be derivation $\delta$ is ill-defined, since you’d have
$$
 \delta(VU) = 1 - (e^{2\pi i\theta})^3U^2 V^2, \quad \delta(e^{2\pi i \theta} UV) = e^{2\pi i \theta} - (e^{2\pi i \theta})^2 U^2V^2.
$$


