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.
Famous quotes containing the words infinite, class and/or property:
“It has no future but itself—
Its infinite contain
Its past—enlightened to perceive
New periods of pain.”
—Emily Dickinson (1830–1886)
“Much of the wisdom of the world is not wisdom, and the most illuminated class of men are no doubt superior to literary fame, and are not writers.”
—Ralph Waldo Emerson (1803–1882)
“The awareness of the all-surpassing importance of social groups is now general property in America.”
—Johan Huizinga (1872–1945)