There exists a canonical action of the group $S^1\times S^1$ on the irrational rotation algebra $A_\theta$, which is the universal C*algebra generated by two unitaries $u$ and $v$ satisfying $uv=e^{i2\pi\theta}vu$, where $\theta$ is an irrational between 0 and 1. That action maps each pair $(z_1,z_2)$, to the automorphism of $A_\theta$ determined by $u\mapsto z_1 u$ and $v\mapsto z_2 v$.
In his 1980 paper, "C$^*$algébres et géométrie différentielle", Alain Connes states that the sub-algebra of the elements of $A_\theta$ whose orbit under that action is smooth consists precisely of those elements of the form $\sum_{(n,m)\in\mathbb{Z}\times\mathbb{Z}}s_{n,m}u^n v^m$, where the complex sequence $(s_{n,m})$ is rapidly decreasing. Since then, many subsequent papers (for example, Bhatt, Inoue & Ogi, "Differential Structures in C*algebras", 2011, Example 6.2.iii) also state this fact, but I have never seen a proof of the harder part of the statement (it is not too hard to prove that the elements of that form do have smooth orbit).
Could anybody help?
This kind of folklore is endemic in NCG. At the risk of immodesty, let me point you to section 2 (specifically proposition 2.18) of this paper, which arose because I needed to use the same folklore but wasn't confident enough to wield it as folklore.