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| In [[mathematics]], there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to [[isomorphism]] there is a unique [[von Neumann algebra]] that is a [[factor (functional analysis)|factor]] of type II<sub>1</sub> and also [[von Neumann algebra#Amenable von Neumann algebras|hyperfinite]]; it is called the '''hyperfinite type II<sub>1</sub> factor'''.
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| There are an uncountable number of other factors of type II<sub>1</sub>. [[Alain Connes|Connes]] proved that the infinite one is also unique.
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| ==Constructions==
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| *The [[von Neumann algebra#Examples|von Neumann group algebra]] of a discrete group with the [[infinite conjugacy class property]] is a factor of type II<sub>1</sub>, and if the group is [[amenable]] and [[countable]] the factor is hyperfinite. There are many groups with these properties, as any [[locally finite group]] is amenable. For example, the von Neumann group algebra of the infinite symmetric group of all permutations of a countable infinite set that fix all but a finite number of elements gives the hyperfinite type II<sub>1</sub> factor.
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| *The hyperfinite type II<sub>1</sub> factor also arises from the [[Crossed_product#Construction|group-measure space construction]] for ergodic free measure-preserving actions of countable amenable groups on probability spaces.
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| *The [[Von_Neumann_algebra#Tensor_products_of_von_Neumann_algebras|
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| infinite tensor product]] of a countable number of factors of type I<sub>''n''</sub> with respect to their tracial states is the hyperfinite type II<sub>1</sub> factor. When ''n''=2, this is also sometimes called the Clifford algebra of an infinite separable Hilbert space.
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| *If ''p'' is any non-zero finite projection in a hyperfinite von Neumann algebra ''A'' of type II, then ''pAp'' is the hyperfinite type II<sub>1</sub> factor. Equivalently the [[Von_Neumann_algebra#Type_II_factors|fundamental group]] of ''A'' is the group of all positive real numbers. This can often be hard to see directly. It is, however, obvious when ''A'' is the infinite tensor product of factors of type I<sub>n</sub>, where n runs over all integers greater than 1 infinitely many times: just take ''p'' [[Von_Neumann_algebra#Projections|equivalent]] to an infinite tensor product of projections ''p''<sub>''n''</sub> on which the tracial state is either 1 or <math> 1- 1/n</math>.
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| ==Properties==
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| The hyperfinite II<sub>1</sub> factor ''R'' is the unique smallest infinite
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| dimensional factor in the following sense: it is contained in any other infinite dimensional factor, and any infinite dimensional factor contained in ''R'' is isomorphic to ''R''.
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| The outer automorphism group of ''R'' is an infinite simple group with countable many conjugacy classes, indexed by pairs consisting of a positive integer ''p'' and a complex ''p''th root of 1.
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| The projections of the hyperfinite II<sub>1</sub> factor form a [[continuous geometry]].
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| ==The infinite hyperfinite type II factor==
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| While there are other factors of [[von Neumann algebra|type II<sub>∞</sub>]], there is a unique [[hyperfinite]] one, up to isomorphism. It consists of those infinite square matrices with entries in the hyperfinite type II<sub>1</sub> factor that define [[bounded operator]]s.
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| ==See also==
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| *[[Subfactor]]s
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| ==References==
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| *A. Connes, [http://links.jstor.org/sici?sici=0003-486X%28197607%292%3A104%3A1%3C73%3ACOIFC%3E2.0.CO%3B2-V ''Classification of Injective Factors''] The Annals of Mathematics 2nd Ser., Vol. 104, No. 1 (Jul., 1976), pp. 73–115
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| *F.J. Murray, J. von Neumann, [http://links.jstor.org/sici?sici=0003-486X%28194310%292%3A44%3A4%3C716%3AOROOI%3E2.0.CO%3B2-O ''On rings of operators IV''] Ann. of Math. (2), 44 (1943) pp. 716–808. This shows that all approximately finite factors of type II<sub>1</sub> are isomorphic.
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| [[Category:Von Neumann algebras]]
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I attempt to keep as toned as possible staying at the gymnasium many times a week. I appreciate my athletics and try to play or see while many a possible. Being wintertime I shall often at Hawthorn suits. Note: I've seen the carnage of fumbling matches at stocktake sales, If you really contemplated purchasing a hobby I do not mind.
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