Group von Neumann algebra ($L(\Gamma )=\lambda (\Gamma {)}^{\prime \prime }$)

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 $\rho (g){\delta }_h={\delta }_{h{g}^{-1}}$.