Group von Neumann algebra

Summary

The group von Neumann algebra $L(G)$ of a locally compact group $G$ is the von Neumann algebra generated by the left translation operators on $L^2(G)$. It is the weak closure of the group algebra acting on $L^2(G)$. For discrete groups it is a main source of examples of factors. When $G$ is a nontrivial discrete group in which every conjugacy class other than $\set{e}$ is infinite, $L(G)$ is a factor of type $\mathrm{II}_1$.