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 religion of the Bible is the best in the world. I see the infinite value of religion. Let it be always encouraged. A world of superstition and folly have grown up around its forms and ceremonies. But the truth in it is one of the deep sentiments in human nature.”
—Rutherford Birchard Hayes (1822–1893)
“Where justice is denied, where poverty is enforced, where ignorance prevails, and where any one class is made to feel that society is in an organized conspiracy to oppress, rob, and degrade them, neither persons nor property will be safe.”
—Frederick Douglass (c. 1817–1895)
“The diversity in the faculties of men, from which the rights of property originate, is not less an insuperable obstacle to a uniformity of interests. The protection of these faculties is the first object of government.”
—James Madison (1751–1836)