In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.[1]: 907
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.[2]
Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed,[1]: 908 and free groups on two generators.[1]: 908
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.
^ abcCite error: The named reference Palmer was invoked but never defined (see the help page).
^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, ISBN 978-3-03719-022-7, MR 2334200. See in particular p. 450: "LΓ is a II1 factor iff Γ is ICC".
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