When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.

What is known about the following question: for which locally compact groups $G$ is $vN(G)$ a factor?

My understanding is that a complete characterisation isn't known - correct me if I'm wrong. What is it that makes the locally compact case so much harder than the discrete case?