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:
“The universe is then one, infinite, immobile.... It is not capable of comprehension and therefore is endless and limitless, and to that extent infinite and indeterminable, and consequently immobile.”
—Giordano Bruno (1548–1600)
“The traveler to the United States will do well ... to prepare himself for the class-consciousness of the natives. This differs from the already familiar English version in being more extreme and based more firmly on the conviction that the class to which the speaker belongs is inherently superior to all others.”
—John Kenneth Galbraith (b. 1908)
“Thieves respect property. They merely wish the property to become their property that they may more perfectly respect it.”
—Gilbert Keith Chesterton (1874–1936)