{{Short description, Unique von Neumann algebra In

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

there is a unique

von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann al ...

that is a

factor Factor (Latin, ) may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, such a factor is a resource used ...

of type II₁ and also hyperfinite; it is called the hyperfinite type II₁ factor. There are an uncountable number of other factors of type II₁. Connes proved that the infinite one is also unique.

Constructions

*The von Neumann group algebra of a discrete group with the infinite conjugacy class property is a factor of type II₁, and if the group is amenable and

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

the factor is hyperfinite. There are many groups with these properties, as any

locally finite group In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studie ...

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 II₁ factor. *The hyperfinite type II₁ factor also arises from the group-measure space construction for ergodic free measure-preserving actions of countable amenable groups on probability spaces. *The infinite tensor product of a countable number of factors of type I_''n'' with respect to their tracial states is the hyperfinite type II₁ factor. When ''n''=2, this is also sometimes called the Clifford algebra of an infinite separable Hilbert space. *If ''p'' is any non-zero finite projection in a hyperfinite von Neumann algebra ''A'' of type II, then ''pAp'' is the hyperfinite type II₁ factor. Equivalently the

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of ''A'' is the group of

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. This can often be hard to see directly. It is, however, obvious when ''A'' is the infinite tensor product of factors of type I_n, where n runs over all integers greater than 1 infinitely many times: just take ''p''

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to an infinite tensor product of projections ''p''_''n'' on which the tracial state is either

1

1- 1/n

Properties

The hyperfinite II₁ factor ''R'' is the unique smallest infinite 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''. 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. The projections of the hyperfinite II₁ factor form a

continuous geometry In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, \dots, \textit, it can be an element of the unit interval ,1 Von Neuman ...

The infinite hyperfinite type II factor

While there are other factors of type II_∞, there is a unique hyperfinite one, up to isomorphism. It consists of those infinite square matrices with entries in the hyperfinite type II₁ factor that define

bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...

References

*A. Connes
''Classification of Injective Factors''
The Annals of Mathematics 2nd Ser., Vol. 104, No. 1 (Jul., 1976), pp. 73–115 *F.J. Murray, J. von Neumann,
''On rings of operators IV''
Ann. of Math. (2), 44 (1943) pp. 716–808. This shows that all approximately finite factors of type II₁ are isomorphic. Von Neumann algebras

Constructions

Properties

The infinite hyperfinite type II factor

See also

References