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

*Question 0.1* (Sakai, Problem 4.4.10 [here][2], 1971): Is every $\mathrm{II}_1$ factor a group von Neumann algebra?  
*Answer* (Connes [here][3], 1975): No, because there is a $\mathrm{II}_1$ factor which is *not* anti-isomorphic to itself (whereas $L\Gamma$ is so).  

*Question 0.2*: Is every $\mathrm{II}_1$ factor anti-isomorphic to itself a group von Neumann algebra?   
*Answer* (Jones, [here][4], 1980): No, because there is a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $L\Gamma$ has so).    

The answers for the next two questions were pointed out by Jiang's comments. 

*Question 0.3* (Remark 5.7 in previous paper): Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism a group von Neumann algebra?  
*Answer* (Ioana [here][5], 2010): No, because there are $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras (see Corollary F, Corollary 10.1 and Remark 10.3).

*Question 0.4*: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomormphic to a group von Neumann algebra (i.e. of the form $(L\Gamma)^t$)?   
*Answer* (Boutonnet [here][6], 2013): No, because there are crossed product von Neumann algebra $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$, with $σ$ a pmp-action of an ICC group $Γ$ on a probability space $(X,μ)$, which are not stably isomorphic to a group von Neumann algebra.   

**Question**: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to a group von Neumann algebra $L \Gamma$ or a crossed product $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$ (as above)?   
[if not, more generally, stably isomorphic to $A⋊Γ$ with $A$ an abelian von Neumann algebra?]  

*Remark*: The existence of a Kac algebra generating a $\mathrm{II}_1$ factor not stably isomorphic to any $A⋊Γ$ (as above) would be very interesting. 

The initial question and then Connes' paper were pointed out to me by [Keshab Chandra Bakshi][7].


  [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-getitem?mr=3077866
  [7]: https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=1197952