# Is the algebra of Schwarz functions on a noncommutative torus the maximal algebra of smooth functions?

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 $$A^{\infty}_{\theta}$$.

My question is: Is $$A^{\infty}_{\theta}$$ also the maximal noncommutative smooth functions on a noncommutative $$2$$-torus, i.e. the maximal subalgebra in $$A_{\theta}$$ such that arbitrary composition of basic derivations could be defined?

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.