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  "body": "\n## Summary\n\nThe group von Neumann algebra $L(G)$ of a [[locally compact group|AHV05ZBCYW]] $G$ is the [[von Neumann algebra|PSFWQVZ0FS]] generated by the left translation operators on $L^2(G)$.\nIt is the weak closure of the group algebra acting on $L^2(G)$.\nFor discrete groups it is a main source of examples of factors.\nWhen $G$ is a nontrivial discrete group in which every conjugacy class other than $\\set{e}$ is infinite, $L(G)$ is a factor of type $\\mathrm{II}_1$.\n\n## Explanations\n\n### L1 human\n\nA group acts on functions on itself by shifting them, and each shift is a unitary operator on the Hilbert space $L^2(G)$.\nTaking all these operators, their linear combinations and their limits in a weak sense gives the group von Neumann algebra.\nFor the integers, Fourier series turn shifting sequences into multiplying functions on the circle by $z^n$.\nSo $L(\\Z)$ is the algebra of bounded measurable functions on the circle, acting by multiplication.\nFor noncommutative groups such as free groups, the algebra is much more mysterious, and its study is a large part of operator algebra theory.\n",
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