Infinite conjugacy class property

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]^: 907

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,^[1]^: 908 and free groups on two generators.^[1]^: 908

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

^ ^a ^b ^c 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, ISBN 9780521366380.
^ Popa, Sorin (2007), "Deformation and rigidity for group actions and von Neumann algebras", International Congress of Mathematicians. Vol. I (PDF), vol. 1, 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".

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[Palmer-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, ISBN 9780521366380.

[2] Popa, Sorin (2007), "Deformation and rigidity for group actions and von Neumann algebras", International Congress of Mathematicians. Vol. I (PDF), vol. 1, 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".

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