Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite.
Does there exist a generalization of this theorem to arbitrary locally compact groups?
A. Connes result from 1976 is that if $G$ is separable and locally compact and $G/G_0$ is amenable, where $G_0$ is the connected component of the identity, then $\mathcal L(G)$ is approximately finite-dimensional. In particular, this occurs when $G$ is solvable separable locally compact, and when $G$ is connected separable locally compact.
There might be newer results, since a lot of work has been done on group von Neumann algebras since then.