Let $\theta$ be a real number. We define $A\_{\theta}$, the algebra of continuous functions on a noncommutative $2$-torus, to be the universal $C^\*$-algebra generated by two generators $U$ and $V$ which satisfy
$$
UV=e^{2\pi i\theta}VU, UU\*=U^\*U=VV^\*=V^\*V=1.
$$


In literature people define smooth functions on a noncommutative torus to be a dense subalgebra $A^{\infty}\_{\theta}$of $A\_{\theta}$ as
$$
A^{\infty}\_{\theta}:=\{\sum\_{r,s\in \mathbb{Z}}a\_{rs}U^rV^s|\{a\_{rs}\}\in \mathcal{S}(\mathbb{Z^2})\}
$$
where $\mathcal{S}(\mathbb{Z^2})$ denotes the space of rapid decreasing functions on $\mathbb{Z}^2$.


I'm not sure if the definition of $A^{\infty}\_{\theta}$ is a convenient choice. In the commutative case, i.e. when $\theta=0$. $A^{\infty}\_{\theta}$ coincides with the algebra of smooth functions on $T^2$ by Fourier transformation. In the noncommutative case, we can still do lots of differential operations on $A^{\infty}\_{\theta}$. For example we have basic derivations on $A^{\infty}\_{\theta}$. This means that elements in $A^{\infty}\_{\theta}$ must be smooth but I wonder if there exist "smooth functions" outside $A^{\infty}\_{\theta}$.



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Is the algebra of Schwarz functions on a noncommutative torus the maximal algebra of smooth functions?
The two basic derivations are infinitesimal generators of an action $\phi$ of $\mathbb{T}^2$ on $A\_\theta$, and the Fourier coefficients of $A \in A\_\theta$ are recovered by $a\_{rs}U^rV^s = \int\_{\mathbb{T}^2} \phi\_{xy}(A) e^{rx + sy} dxdy$. This shows that the $a\_{rs}$ are bounded by $\|A\|$. The two generators act on the Fourier coefficients by multiplication by $r$ and $s$, so if $A$ is in the domain of compositions of basic derivations then the values $r^k s^l a\_{rs}$ must be uniformly bounded, for each $k$ and $l$. I may be missing some factors of $2\pi i$, etc., but that's the idea. This is covered in Section 5.5 of my book *Mathematical Quantization*.


