The group von Neumann algebra $LG$ is a factor if and only if the group $G$ is ICC (i.e. infinite conjugacy class property). 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 of S. Sakai, 1971). A. Connes answered negatively in this paper (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 (1980) by building a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $LG$ has so). Then:

Question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism of the form $LG$?
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.