# Is the von neumann algebra of locally compact amenable group hyperfinite?

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.
• That's surprising (probably he had made a mistake) because if you take a separable but non-second-countable group like $\{0,1\}^{[0,1]}$ then there is a problem with the definition of hyperfiniteness (the usual conditions that we use for separably acting von Neumann algebras are no longer equivalent). – Tomasz Kania Jun 17 '15 at 14:40