Infinite conjugacy class property
The von Neumann group algebra of a group is a factor if and only if the group has the infinite conjugacy class property. It will then be, provided the group is nontrivial, of type II1, i.e. it will possess a unique, faithful, tracial state.
In abelian groups, every conjugacy class consists of only one element, so icc groups are, in a way, as far from being abelian as possible.
- Palmer, Theodore W. (2001), Banach Algebras and the General Theory of *-Algebras, Volume 2, Encyclopedia of mathematics and its applications, 79, Cambridge University Press, p. 907, ISBN 9780521366380.
- Popa, Sorin (2007), "Deformation and rigidity for group actions and von Neumann algebras", International Congress of Mathematicians. Vol. I (PDF), Eur. Math. Soc., Zürich, pp. 445–477, doi:10.4171/022-1/18, MR 2334200. See in particular p. 450: "LΓ is a II1 factor iff Γ is ICC".
- Palmer (2001), p. 908.