Noncommutative torus

Example

Let $\theta \in \R$. The noncommutative torus $A_\theta$ is the universal C*-algebra generated by two unitaries $U$ and $V$ with $$V U = e^{2 \pi i \theta} U V.$$

Concrete check: on $L^2(\mathbb{T})$, with $\mathbb{T}$ the unit circle and its arc length measure, let $(U f)(z) = z f(z)$ and $(V f)(z) = f(e^{2 \pi i \theta} z)$. Both are unitary. Then $(V U f)(z) = e^{2 \pi i \theta} z f(e^{2 \pi i \theta} z)$ and $(U V f)(z) = z f(e^{2 \pi i \theta} z)$, so $V U = e^{2 \pi i \theta} U V$. For irrational $\theta$, the C*-algebra generated by these $U$ and $V$ is isomorphic to $A_\theta$.