Infinite_conjugacy_class_property

Infinite conjugacy class property

Infinite conjugacy class property

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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.[1]

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.[2]

Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed,[3] and free groups on two generators.[3]

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.

References

  1. [1]
    Palmer, Theodore W. (2001), Banach Algebras and the General Theory of *-Algebras, Volume 2, Encyclopedia of mathematics and its applications, vol. 79, Cambridge University Press, p. 907, ISBN 9780521366380.
  2. [2]
    Popa, Sorin (2007), "Deformation and rigidity for group actions and von Neumann algebras", International Congress of Mathematicians. Vol. I (PDF), Eur. Math. Soc., Zürich, pp. 445–477, doi:10.4171/022-1/18, ISBN 978-3-03719-022-7, MR 2334200. See in particular p. 450: "LΓ is a II1 factor iff Γ is ICC".
  3. [3]
    Palmer (2001), p. 908.


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