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Infinite conjugacy class property
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In [[mathematics]], a [[group (mathematics)|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 [[Infinity|infinite]].{{r|Palmer|p=907}} The [[von Neumann group algebra]] of a group is a [[factor (functional analysis)|factor]] if and only if the group has the infinite conjugacy class property. It will then be, provided the group is nontrivial, of type ''II<sub>1</sub>'', i.e. it will possess a unique, faithful, tracial state.<ref>{{citation | last = Popa | first = Sorin | contribution = Deformation and rigidity for group actions and von Neumann algebras | doi = 10.4171/022-1/18 | mr = 2334200 | pages = 445–477 | publisher = Eur. Math. Soc., Zürich | title = International Congress of Mathematicians. Vol. I | url = http://www.icm2006.org/proceedings/Vol_I/22.pdf | year = 2007| volume = 1 | isbn = 978-3-03719-022-7 }}. See in particular p. 450: "''L''Γ is a II<sub>1</sub> factor iff Γ is ICC".</ref> Examples of ICC groups are the group of [[permutation]]s of an infinite set that leave all but a finite subset of elements fixed,{{r|Palmer|p=908}} and [[free group]]s on two generators.{{r|Palmer|p=908}} In [[abelian group]]s, every conjugacy class consists of only one element, so ICC groups are, in a way, as far from being abelian as possible. ==References== {{reflist|refs= <ref name=Palmer>{{citation|title=Banach Algebras and the General Theory of *-Algebras, Volume 2|series=Encyclopedia of mathematics and its applications|volume=79|first=Theodore W.|last=Palmer|publisher=Cambridge University Press|year=2001|isbn=9780521366380|url=https://books.google.com/books?id=zn-iZNNTb-AC}}.</ref> }} {{DEFAULTSORT:Infinite Conjugacy Class Property}} [[Category:Infinite group theory]] [[Category:Properties of groups]] {{algebra-stub}}
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