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Lance's theorem asserts that a discrete group $G$ is amenable if and only if the reduced and full groups C*-algebras coincide. The group von Neumann algebra is the weak closure of the reduced group C*-algebra concretely represented on $\ell_2(G)$. The definition of the full group C*-algebra is abstract as we range through all possible repsresntations of the algebraic group algebra so we can't mimick the definition of the group von Neumann algebra in this case.

Does the full group C*-algebra canonically embed into some finite von Neumann algebra like the reduced group C*-algebra does?

If so, what is the relation (if any) to the group von Neumann algebra of $G$?

Tomasz Kania
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2 Answers2

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I think the answer is no. Glimm proved that every group that has a unitary representation whose image generates a type II factor, also has a unitary representation that generates a type III factor. When you look at the full group C$^*$-algebra all representations are on equal footing, so you cannot embed it canonically in an environment where there is a trace.

Martin Argerami
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  • Thanks Martin. What worries me a little is that you make no reference to amenability whatsoever. In this case of course, the reduced and full C*-algebras are the same, yet by Glimm's result there are still representations which generate type III factors. – Tomasz Kania Jun 16 '15 at 13:33
  • Yes, I actually waited for a while to post my answer because of that, and my last sentence vaguely tried to address it. I think that the point is simply that when the group is amenable all its faithful representations can be made to live in the same algebra and are thus isomorphic (weather they live in a II $_1$ factor or a III factor), which is not the case for non-amenable. – Martin Argerami Jun 16 '15 at 15:59
  • I think I don't get it. There are still type III factors lurking behind amenable groups. The point is that they contribute nothing to the norm(s) on the algebraic group algebra. – Tomasz Kania Jun 16 '15 at 22:01
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I'm pretty sure that the answer to this 8-yrs-old question is known to OP by now. The aim of this post is to provide a (partial) answer & a reference to the newcomers.

$C^*(G)$ embeds into a finite von Neumann algebra iff the FF-representations (unitary factor representations of finite type) separates the points of $G$. This follows simply from the factor decomposition of von Neumann algebras. If so, for example, we can choose $\bigoplus \pi(G)''$ as the vN-algebra that $C^*(G)$ embeds in, where $\pi$ ranges through all FF representations of $G$.

Every SIN group with a neighborhood system of compact open conjugation-invariant neighborhoods has a separating family of FF representations (17.3.6 in Dixmier's C*-algebra book). Conversely, if $G$ is a compactly generated LCG and has a separating family of FF representations, then $G$ is a SIN group with a neighborhood system of compact open conjugation-invariant neighborhoods (17.3.7 in Dixmier's book).

edit: There's a flaw in the argument above. $C^*(G)$ embeds into a finite von Neumann algebra iff the FF representations separates the points of $\mathbf{C^*(G)}$ (and not merely $G$).

Due to a result by Kirchberg, Connes embedding problem has a positive answer iff $C^*(SL_2(\mathbb{Z})\times SL_2(\mathbb{Z}))$ embeds into a finite vN-algebra. The group $SL_2(\mathbb{Z})\times SL_2(\mathbb{Z})$ is discrete, so a SIN group as above, so has a separating family of FF representations; however, it is an open problem (to the best of my knowledge) that FF representations separates the points of $C^*(SL_2(\mathbb{Z})\times SL_2(\mathbb{Z}))$.

Onur Oktay
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  • Pls. also see the article in the link that claims a negative solution to Connes embedding problem https://doi.org/10.1090/bull/1768 – Onur Oktay Jun 05 '24 at 17:51