---
tag: PZ97S3QQF1
kind: example
name: Noncommutative torus
msc: [46L05, 46L87, 58B34]
sources:
  - url: https://en.wikipedia.org/wiki/Noncommutative_torus
same_as:
  wikidata: Q3532264
  wikipedia: 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 $A_\theta$.

## Remarks

For $\theta = 0$, $U$ and $V$ commute, and $A_0 \cong C(\mathbb{T}^2)$, the continuous functions on the ordinary torus; this explains the name.
For irrational $\theta$, the algebra $A_\theta$ is simple and has a unique tracial state; it is also called the irrational rotation algebra.
It is the [[crossed product|8HG8TS46A6]] $C(\mathbb{T}) \rtimes \Z$ for the action of $\Z$ on $\mathbb{T}$ by rotation through the angle $2 \pi \theta$.
For irrational $\theta$ and $\theta'$, $A_\theta \cong A_{\theta'}$ if and only if $\theta' \equiv \pm \theta \pmod{1}$.

## Explanations

### L1 human

Functions on the ordinary torus are generated by the two coordinate functions $z_1$ and $z_2$, which commute.
The noncommutative torus keeps two generators but lets them commute only up to a fixed phase $e^{2 \pi i \theta}$.
It is a basic example of a "noncommutative space" in noncommutative geometry.