= Group von Neumann algebra
{title2=$L(\Gamma)=\lambda(\Gamma)''$}
For a discrete <group>, its <group von Neumann algebra> is generated on $\ell^2(\Gamma)$ by left translations $\lambda(g)\delta_h=\delta_{gh}$. The <vector> $\delta_e$ is cyclic and separating, and its <vector> functional is tracial. Its <modular conjugation> acts by $J\delta_g=\delta_{g^{-1}}$ with conjugated scalar coefficients. The commutant is the algebra generated by right translations $\rho(g)\delta_h=\delta_{hg^{-1}}$.
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