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  "body": "\n## Definition\n\nLet $G$ be a second countable [[locally compact group|AHV05ZBCYW]].\nA [[unitary representation|74BPYJNZTM]] $\\pi$ of $G$ on a [[Hilbert space|HM2HHE3J51]] $\\mathcal{H}$ is *mixing* if $\\ip{\\pi(g)\\xi}{\\eta} \\to 0$ as $g \\to \\infty$, for all $\\xi, \\eta \\in \\mathcal{H}$.\nHere $g \\to \\infty$ means that $g$ leaves every [[compact|0TVJZYB6SH]] subset of $G$.\nThe representation $\\pi$ *almost has invariant vectors* if the following holds.\nFor every compact $Q \\subseteq G$ and $\\varepsilon \u003e 0$, there is a unit vector $\\xi$ with $\\norm{\\pi(g)\\xi - \\xi} \u003c \\varepsilon$ for all $g \\in Q$.\nThe group $G$ has the *Haagerup property* if it has a mixing unitary representation that almost has invariant vectors.\n\n## Remarks\n\nExample: $G = \\Z$ and the regular representation on $\\ell^2(\\Z)$.\nFor finitely supported $\\xi$ and $\\eta$, the coefficient $\\ip{\\pi(n)\\xi}{\\eta}$ is $0$ once $\\abs{n}$ is large, and by density the representation is mixing.\nThe unit vectors $N^{-1/2} \\mathbf{1}_{\\set{0, \\ldots, N-1}}$ are almost invariant for large $N$.\n\nNon-example: $\\mathrm{SL}_3(\\Z)$ does not have the Haagerup property.\nIt has [[property (T)|GTFZ277047]], so a representation that almost has invariant vectors has a nonzero invariant vector $\\xi$.\nThen $\\ip{\\pi(g)\\xi}{\\xi} = \\norm{\\xi}^2$ for all $g$, which does not tend to $0$ on the infinite group.\n\nEquivalently, $G$ has a proper continuous conditionally negative definite function $\\psi \\colon G \\to [0, \\infty)$.\nAnother equivalent form: $G$ has a continuous affine isometric action on a Hilbert space that is metrically proper.\nAmenable groups, free groups and $\\mathrm{SL}_2(\\R)$ have the Haagerup property.\n",
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    "name": "Haagerup property",
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