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:
“What means the fact—which is so common, so universal—that some soul that has lost all hope for itself can inspire in another listening soul an infinite confidence in it, even while it is expressing its despair?”
—Henry David Thoreau (1817–1862)
“Women ... are completely alone, though they were born and bred upon this soil, as if they belonged to another class in creation.”
—“Jennie June” Croly 1829–1901, U.S. founder of the woman’s club movement, journalist, author, editor. F, Demorest’s Illustrated Monthly Mirror of Fashions, pp. 363-4 (December 1870)
“No man is by nature the property of another. The defendant is, therefore, by nature free.”
—Samuel Johnson (1709–1784)