The group von Neumann algebra $LG$ is a factor if and only if the group $G$ is ICC (i.e. [infinite conjugacy class property][1]). Moreover if $G$ is nontrivial then $LG$ is a $\mathrm{II}_1$ factor.    

The initial question is to ask whether every $\mathrm{II}_1$ factor is of this form (Problem 4.4.10 in [this book][2] of S. Sakai, 1971). A. Connes answered negatively in [this paper][3] (1975) by building a $\mathrm{II}_1$ factor which is *not* anti-isomorphic to itself (whereas $LG$ is so). This leads to ask whether every $\mathrm{II}_1$ factor anti-isomorphic to itself is of that form. V. Jones answered negatively in [this paper][4] (1980) by building a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $LG$ has so). **Edit** (after Jiang comment): this leads to ask whether a $\mathrm{II}_1$ factor with an involutory antiautomorphism is of that form. A. Ioana answered negatively in [this paper][5] (2010) by building $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras. Then

**Question**: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism of the form $(LG)^t$?    
  
In other words: Does a $\mathrm{II}_1$ factor with an involutory antiautomorphism admit an amplification which is a group von Neumann algebra?
If not, it there still an abstract characterization of such $\mathrm{II}_1$ factors?

The initial question and then Connes' paper were pointed out to me by [Keshab Chandra Bakshi][6].


  [1]: https://en.wikipedia.org/wiki/Infinite_conjugacy_class_property
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=0442701
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=377534
  [4]: https://mathscinet.ams.org/mathscinet-getitem?mr=585235
  [5]: https://mathscinet.ams.org/mathscinet-getitem?mr=2813341
  [6]: https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=1197952