Let $G$ be a discrete group with identity element $e$ such that every [conjugacy class](/page/Conjugacy%20Class) $\{hgh^{-1}:h\in G\}$ with $g\ne e$ is infinite. Let $\ell^2(G)$ be the [Hilbert space](/page/Hilbert%20Space) with canonical [orthonormal basis](/page/Orthonormal%20Basis) $(\delta_g)_{g\in G}$. Let $\lambda:G\to\mathcal{L}(\ell^2(G))$ be the left regular representation, $\lambda_h\delta_g=\delta_{hg}$, and let $\rho:G\to\mathcal{L}(\ell^2(G))$ be the right regular representation, $\rho_h\delta_g=\delta_{g h^{-1}}$. Define the group von Neumann algebra $L(G)$ by $L(G):=\rho(G)'$, equivalently the strong operator closure of the unital $*$-algebra generated by $\lambda(G)$. Then $L(G)$ is a factor.