Noncommutative two-torus

For $\theta\in\mathbb R$ , the noncommutative two-torus $A_\theta$ is the universal unital $C^*$ -algebra generated by unitaries $u,v$ subject to

vu=e^{2\pi i\theta}uv.

For irrational $\theta$ , it is also called the irrational rotation algebra.

Construction from a circle rotation

For $T_\theta(x)=x+\theta$ , let $\alpha(f)=f\circ T_\theta^{-1}$ on $C(\mathbb T)$ , and let $v(x)=e^{2\pi ix}$ . The covariance relation $uvu^*=\alpha(v)=e^{-2\pi i\theta}v$ gives the defining commutation rule. Thus

A_\theta\cong C(\mathbb T)\rtimes_\alpha\mathbb Z.

The group $\mathbb Z$ is amenable, so full and reduced crossed products agree.

Completions and the commutative value

For $\theta=0$ , the generators commute and $A_0\cong C(\mathbb T^2)$ . The von Neumann algebra $L^\infty(\mathbb T)\rtimes\mathbb Z$ is a different completion, using the Haar tracial representation. For irrational $\theta$ , essential freeness and ergodicity make it a type $\mathrm{II}_1$ factor; it is the hyperfinite $\mathrm{II}_1$ factor.

References

Alain Connes, Noncommutative Geometry, 1994, Introduction and Chapter II, noncommutative tori.