Smooth noncommutative torus

The smooth noncommutative torus $A_\theta^\infty$ is the subalgebra of the rotation algebra $A_\theta$ consisting of series

\sum_{m,n\in\mathbb Z}c_{m,n}v^mu^n

whose coefficients are rapidly decreasing: for every integer $N\geq0$ ,

\sup_{m,n}(1+|m|+|n|)^N|c_{m,n}|<\infty.

Here $vu=e^{2\pi i\theta}uv$ . The series converge in the $C^*$ -norm and define a dense $*$ -subalgebra.

Smoothness and differentiation

Multiplication is twisted convolution of coefficients, which preserves rapid decrease. The canonical derivations multiply coefficients by $2\pi im$ and $2\pi in$ , respectively, and also preserve rapid decrease. At $\theta=0$ , this algebra is $C^\infty(\mathbb T^2)$ .

References

Alain Connes, C-algebras and differential geometry*, English translation of the 1980 note, pp. 4–5 (torus action, derivations, and rapid-decay series).