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  "body": "\n## Example\n\nLet $\\theta \\in \\R$.\nThe *noncommutative torus* $A_\\theta$ is the universal C*-algebra generated by two unitaries $U$ and $V$ with\n$$V U = e^{2 \\pi i \\theta} U V.$$\n\nConcrete 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 $(V f)(z) = f(e^{2 \\pi i \\theta} z)$.\nBoth are unitary.\nThen $(V U f)(z) = e^{2 \\pi i \\theta} z f(e^{2 \\pi i \\theta} z)$ and $(U V f)(z) = z f(e^{2 \\pi i \\theta} z)$, so $V U = e^{2 \\pi i \\theta} U V$.\nFor irrational $\\theta$, the C\\*-algebra generated by these $U$ and $V$ is isomorphic to $A_\\theta$.\n\n## Remarks\n\nFor $\\theta = 0$, $U$ and $V$ commute, and $A_0 \\cong C(\\mathbb{T}^2)$, the continuous functions on the ordinary torus; this explains the name.\nFor irrational $\\theta$, the algebra $A_\\theta$ is simple and has a unique tracial state; it is also called the irrational rotation algebra.\nIt is the [[crossed product|8HG8TS46A6]] $C(\\mathbb{T}) \\rtimes \\Z$ for the action of $\\Z$ on $\\mathbb{T}$ by rotation through the angle $2 \\pi \\theta$.\nFor irrational $\\theta$ and $\\theta'$, $A_\\theta \\cong A_{\\theta'}$ if and only if $\\theta' \\equiv \\pm \\theta \\pmod{1}$.\n\n## Explanations\n\n### L1 human\n\nFunctions on the ordinary torus are generated by the two coordinate functions $z_1$ and $z_2$, which commute.\nThe noncommutative torus keeps two generators but lets them commute only up to a fixed phase $e^{2 \\pi i \\theta}$.\nIt is a basic example of a \"noncommutative space\" in noncommutative geometry.\n",
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