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:
“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (1899–1986)
“You see, after the war—and don’t forget it lasted a hundred years—thousands of us went from door to door, asking for honest work, and we were whipped for begging. The ruling class didn’t say, “Work or starve.” They said “Starve, for you shall not work.””
—Sonya Levien (1895–1960)
“When a strong man, fully armed, guards his castle, his property is safe. But when one stronger than he attacks him and overpowers him, he takes away his armor in which he trusted and divides his plunder.”
—Bible: New Testament, Luke 11:21.22.