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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?

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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.

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    $\begingroup$ Are you sure that Connes was dealing with separable (rather than second-countable) groups? $\endgroup$ – Tomasz Kania Jun 16 '15 at 22:04
  • $\begingroup$ I copied that statement almost verbatim from his paper; he definitely uses the word "separable". Like I said, I don't know about newer stuff. $\endgroup$ – Martin Argerami Jun 17 '15 at 8:12
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    $\begingroup$ 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). $\endgroup$ – Tomasz Kania Jun 17 '15 at 14:40

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