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. 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.
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.
Examples for icc groups are free groups on at least two generators, or, more generally, nontrivial free products.
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