2
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.