mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...

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 algeb ...

that is a

factor Factor, a Latin word meaning "who/which acts", 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, suc ...

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 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 algebra ...

of a discrete group with the

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. The von Neumann group algebra of a group is a factor if and only if th ...

is a factor of type II₁, and if the group is amenable and

countable In mathematics, a set is countable if either it is 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 from it into the natural numbers; ...

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 studied. T ...

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

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

of ''A'' is the group of

positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...

. 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''

equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equivale ...

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/math>. Von Ne ...

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 s ...

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