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)
“There is ... a class of fancies, of exquisite delicacy, which are not thoughts, and to which, as yet, I have found it absolutely impossible to adapt language.... Now, so entire is my faith in the power of words, that at times, I have believed it possible to embody even the evanescence of fancies such as I have attempted to describe.”
—Edgar Allan Poe (1809–1849)
“No man acquires property without acquiring with it a little arithmetic, also.”
—Ralph Waldo Emerson (1803–1882)