Von Neumann algebra

A subalgebra of the algebra of bounded linear operators on a Hilbert space that is self-adjoint (that is, contains together with every operator its adjoint operator ) and that coincides with its bicommutant (that is, it contains all operators that commute with every operator commuting with all operators in ). These algebras were introduced by J. von Neumann . According to a theorem of von Neumann, a self-adjoint subalgebra is a von Neumann algebra if and only if (or its unit ball) is closed in the weak, strong, ultraweak, or ultrastrong operator topology (the uniform operator topology does not suffice). A given symmetric Banach algebra (cf. also Symmetric algebra) is isometrically isomorphic to some von Neumann algebra if and only if it is a -algebra isometric to some dual space; the Banach space for which is uniquely determined up to an isometric isomorphism and can be identified with the space of ultraweakly-continuous linear forms on the von Neumann algebra isometrically isometric to ; this space is denoted by and is called the pre-dual of . Such symmetric Banach algebras are called -algebras. Let be a von Neumann algebra on a Hilbert space , its commutator, its centre, a projection belonging to , and a projection belonging to . The subspace is invariant under , and the family of operators from restricted to forms a von Neumann algebra in , which is denoted by and is called the induced algebra, while the mapping is called the induced mapping of onto ; the family of bounded operators of the form , , on the subspace forms a von Neumann algebra in , which is called reduced. If , then the reduced and the induced von Neumann algebras are the same. An isometric isomorphism of a von Neumann algebra is said to be algebraic; a von Neumann algebra on a Hilbert space is said to be spatially isomorphic to a von Neumann algebra on a space if there exists a unitary operator mapping onto and such that . The intersection of any family of von Neumann algebras on a given Hilbert space is a von Neumann algebra; the smallest von Neumann algebra containing a given set is said to be the von Neumann algebra generated by the set . Let , , be Hilbert spaces, their direct sum, a von Neumann algebra on , and the von Neumann algebra on generated by those operators in for which every is invariant under and the restriction of to lies in ; this von Neumann algebra is called the direct product of the and is denoted by . The operations of forming the tensor product, both finite and infinite, are also defined for von Neumann algebras. A von Neumann algebra is called a factor if its centre consists of multiples of the identity.

Let be a von Neumann algebra and the set of its positive operators. A weight on is an additive mapping from into that is homogeneous under multiplication by positive numbers. A weight is called a trace if for all and all unitary operators in . A trace is said to be finite if for all ; semi-finite if for any the quantity is the least upper bound of the numbers of the form , where and ; exact if , , implies ; normal if for any increasing family of elements in with least upper bound the quantity is the least upper bound of the numbers . A von Neumann algebra is called finite if there is a family of normal finite traces on separating the points of ; properly infinite if there are no non-zero finite traces on ; semi-finite if there is an exact normal semi-finite trace on ; and purely infinite, or an algebra of type , if there are no non-zero normal semi-finite traces on . A von Neumann algebra is called discrete, or of type , if it is algebraically isomorphic to a von Neumann algebra with a commutative commutant; such an algebra is semi-finite. A von Neumann algebra is called continuous if for any non-zero central projection the von Neumann algebra is not discrete. A continuous semi-finite algebra is said to be of type . A finite algebra of type is said to be of type ; a properly infinite algebra of type is said to be of type . Whether a von Neumann algebra belongs to a definite type is equivalent to the fact that its commutant belongs to the same type, but the commutant of a finite von Neumann algebra need not be a finite von Neumann algebra.

Let be a von Neumann algebra, and projections belonging to . Then and are called equivalent, , if there is an element such that and . One writes if there is a projection such that and ; the relation is a partial order. A classification of von Neumann algebras according to type can be carried out in terms of this relation; in particular: A projection is called finite if , , implies ; a von Neumann algebra is finite if and only if the identity projection is finite, and semi-finite if and only if the least upper bound of the family of finite projections is the identity projection.

A von Neumann algebra is semi-finite if and only if it can be realized as the left von Neumann algebra of a certain Hilbert algebra; the elements of the latter are those for which , where is an exact normal semi-finite trace on . For algebras of type the corresponding realization can be obtained by means of generalized Hilbert algebras and weights on von Neumann algebras.

Let be fixed Hilbert spaces of dimension , , let be a Borel space, let be a positive measure on , let be a partition of into disjoint measurable subsets, let be the Hilbert space of square-summable -measurable mappings of into , let

and let for . If , then , where . Let for . A mapping , where is a continuous linear operator on the Hilbert space , is called a measurable field of operators if for any the function is measurable on every set . If is a measurable field of operators and the function is essentially bounded on , then for every there is a unit vector such that -almost everywhere. The mapping defined by for all is a bounded linear operator on , and

Such an operator on is called decomposable. Suppose that for any a von Neumann algebra is defined on ; the mapping is called a measurable field of von Neumann algebras if there exists a sequence of measurable fields of operators such that for any the von Neumann algebra is generated by the operators . The set of all decomposable operators on such that for every is a von Neumann algebra in . It is denoted by

and is called the direct integral of the von Neumann algebras over . Every von Neumann algebra on a separable Hilbert space is isomorphic to a direct integral of factors. An arbitrary von Neumann algebra has an algebraic decomposition, and this is why the theory of factors is of interest for the general theory of von Neumann algebras.

Von Neumann algebras arise naturally in problems connected with operators on a Hilbert space and have numerous applications in operator theory itself and in the representation theory of groups and algebras, as well as in the theory of dynamical systems, statistical physics and quantum field theory.

References

[1a]	F.J. Murray, J. von Neumann, "On rings of operators" Ann. of Math. (2) , 37 (1936) pp. 116–229
[1b]	F.J. Murray, J. von Neumann, "On rings of operators II" Trans. Amer. Math. Soc. , 41 (1937) pp. 208–248
[2]	F.J. Murray, J. von Neumann, "On rings of operators IV" Ann. of Math. (2) , 44 (1943) pp. 716–808
[3]	J. Dixmier, "Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann" , Gauthier-Villars (1957)
[4]	J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
[5]	S. Sakai, "-algebras and -algebras" , Springer (1971)
[6a]	J. von Neumann, "On infinite direct products" Compos. Math. , 6 (1938) pp. 1–77
[6b]	J. von Neumann, "On rings of operators III" Ann. of Math. (2) , 41 (1940) pp. 94–161
[7]	A. Guichardet, "Produits tensoriels infinis et réprésentations des rélations d'anticommutation" Ann. Sci. Ecole Norm. Sup. , 83 (1966) pp. 1–52
[8]	M. Takesaki, "Tomita's theory of modular Hilbert algebras and its applications" , Lect. notes in math. , 128 , Springer (1970)
[9]	L. Zsidó, "Topological decompositions of operator algebras" A. Salam (ed.) , Global analysis and its applications (Trieste, 1972) , 3 , IAEA (1974) pp. 305–308
[10]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)

Comments

In terms of the majorization relation , defined above, the types of von Neumann algebras are as follows. A von Neumann algebra is of type if every non-zero central projection in majorizes a non-zero Abelian projection. (An Abelian projection is a projection such that is Abelian.) If there are no non-zero finite projections in , i.e. is purely infinite, then it is of type . If has no non-zero Abelian projections and if every non-zero central projection in majorizes a non-zero finite projection of , then is of type . If is finite and of type , then it is of type . If is of type and has no non-zero central finite projections, then is of type . Every von Neumann algebra is uniquely decomposable into the direct sum of von Neumann algebras of types , , , , and a factor is hence of one of these types.

A factor of type is isomorphic to for some Hilbert space . A factor of type is the algebra of -matrices over .

A von Neumann algebra or factor is hyperfinite if it is generated by an ascending sequence of finite factors, i.e. matrix algebras. There is just one hyperfinite type factor and one hyperfinite factor (up to isomorphism), [a7]. For more details on types and factors, e.g. types , , and finer classification results, cf. [a1], [a4], [a5], [a7]–[a9].

Part of a recent breakthrough in knot theory came from work on the classification of subfactors; cf. (the editorial comments to) Knot theory and references given there, as well as [a3], [a6].

References

[a1]	O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , I-II , Springer (1979)
[a2]	R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , 1–2 , Acad. Press (1986)
[a3]	V.F.R. Jones, "A new knot polynomial and von Neumann algebras" Notices Amer. Math. Soc. , 33 (1986) pp. 219–225
[a4]	V.F.R. Jones, "Subfactors and related topics" D.E. Evans (ed.) M. Takesaki (ed.) , Operator Algebras and Appl. , 2 , Cambridge Univ. Press (1988) pp. 103–118
[a5]	A. Ocneanu, "Quantized groups, strong algebras, and Galois theory for algebras" D.E. Evans (ed.) M. Takesaki (ed.) , Operator Algebras and Appl. , 2 , Cambridge Univ. Press (1988) pp. 119–172
[a6]	P. de la Harpe, M. Kervaire, C. Weber, "On the Jones polynomial" Enseignement Math. , 32 (1986) pp. 271–335
[a7]	A. Connes, "Classification of injective factors" Ann. of Math. , 104 (1976) pp. 73–115
[a8]	A. Connes, M. Takesaki, "The flow of weights on factors on type III" Tohoku Math. J. , 29 (1977) pp. 473–573
[a9]	A. Connes, "Une classification des facteurs de type III" Ann. Sci. École Norm. Sup. , 6 (1973) pp. 133–252