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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\\mu$.\nFor $g \\in G$, let $\\lambda(g)$ be the unitary operator on $L^2(G, \\mu)$ given by $(\\lambda(g)\\xi)(x) = \\xi(g^{-1}x)$, the left regular representation.\nThe *group von Neumann algebra* of $G$ is\n$$L(G) = \\set{\\lambda(g) : g \\in G}'',$$\nthe double [[commutant|1VC7EJ1ZBZ]] of the set of operators $\\lambda(g)$ in the bounded operators on $L^2(G)$.\n\n## Remarks\n\nBy the von Neumann bicommutant theorem, $L(G)$ is the closure in the [[strong operator topology|KHJ2G2R5G7]] of the linear span of the operators $\\lambda(g)$, and it is a [[von Neumann algebra|PSFWQVZ0FS]].\n\nExample: $G = \\Z$.\nFourier series identify $\\ell^2(\\Z)$ with $L^2(\\mathbb{T})$, and $\\lambda(n)$ with multiplication by $z^n$.\nThe commutant of these multiplications is the algebra of multiplications by functions in $L^\\infty(\\mathbb{T})$, which is its own commutant, so $L(\\Z) \\cong L^\\infty(\\mathbb{T})$.\n\nNon-example: the reduced group C\\*-algebra $C^*_r(\\Z) \\cong C(\\mathbb{T})$, acting by multiplication on $L^2(\\mathbb{T})$, is not a von Neumann algebra.\nIts double commutant is $L^\\infty(\\mathbb{T})$, which is strictly larger than $C(\\mathbb{T})$.\n\nFor a discrete group $G$, the vector $\\delta_e$ gives a faithful normal trace $\\tau(a) = \\ip{a\\delta_e}{\\delta_e}$ on $L(G)$.\nIf $G$ is a nontrivial discrete group and every conjugacy class of $G$ other than $\\set{e}$ is infinite, then $L(G)$ is a factor of type $\\mathrm{II}_1$.\n",
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