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:
“Vanity is as advantageous to a government as pride is dangerous. To be convinced of this we need only represent, on the one hand, the numberless benefits which result from vanity, as industry, the arts, fashions, politeness, and taste; and on the other, the infinite evils which spring from the pride of certain nations, a laziness, poverty, a total neglect of everything.”
—Charles Louis de Secondat Montesquieu (1689–1755)
“Criminals are never very amusing. It’s because they’re failures. Those who make real money aren’t counted as criminals. This is a class distinction, not an ethical problem.”
—Orson Welles (1915–1985)
“The rights and interests of the laboring man will be protected and cared for, not by the labor agitators, but by the Christian men to whom God in His infinite wisdom has given control of the property interests of the country.”
—George Baer (1842–1914)