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, "![]() |
[5] | S. Sakai, "![]() ![]() |
[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 |
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