A von Neumann algebra on a Hilbert space bicommutant algebra of bounded
operators from
A non-trivial corollary of the so-called bicommutant theorem says that a nondegenerate algebra iff it is strongly closed . This
is further equivalent to a number of other analytic properties of algebras .
In some literature, the assumption of identity ) is added
to the hypotheses of this equivalence though, strictly speaking, the result holds
in the somewhat more general case that
One can easily show that every von Neumann algebra is a W * -algebraalgebra
as a C * -algebraBanach space
For a free group group
von Neumann algebra group
representation .
An operator algebra weight may take the value infinity on positive
elements .
An injective von Neumann algebra bounded
projection from norms for every linear
map completely positive maps .
OpenAI (2026a) proved that the free group von Neumann algebras isomorphic for C * -algebrafreely
independent semicircular families
of at least two elements.
OpenAI (2026b) separately announced that every type factor with separable predual is generated by one operator ,
equivalently by two self-adjoint operators .
OpenAI (2026c) also announced that sufficiently close unital von Neumann algebras
on a common complex Hilbert space are conjugate
by a unitary arbitrarily close to the identity ,
uniformly over the algebras and representations. The
closeness is measured by the Hausdorff distance
between their operator norm unit
balls .
See also Bicentralizer ,
Bicommutant ,
Bicommutant Theorem ,
Bounded
Hochschild Cohomology ,
C * -Algebra,
Commutant ,
Connes's
Rigidity Conjecture ,
Continuous Core ,
Free
Entropy ,
Fuglede-Kadison Determinant ,
Group von Neumann Algebra ,
Irrational
Rotation Algebra ,
Kadison-Kastler Conjecture ,
Kadison-Kastler Distance ,
Kadison-Ringrose
Cohomology Conjecture ,
Near Inclusion
of Operator Algebras ,
Nondegenerate
Operator Action ,
Operator Algebra State ,
Operator Paving ,
Predual ,
von Neumann Factor ,
W * -Algebra
Portions of this entry contributed by Christopher
Stover
Portions of this entry contributed by Mohammad
Sal Moslehian
Explore with Wolfram|Alpha
References Blackadar, B. "Operator Algebras: Theory of http://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf . Dixmier,
J. Von Neumann Algebras. Amsterdam, Netherlands: North-Holland, 1981. Iyanaga,
S. and Kawada, Y. (Eds.). "Von Neumann Algebras." §430 in Encyclopedic
Dictionary of Mathematics. Kadison, R. V. and Ringrose, J. R. Fundamentals
of the Theory of Operator Algebras, Vol. 2: Advanced Theory. Kadison, R. V. and Ringrose, J. R. Fundamentals
of the Theory of Operator Algebras, Vol. 1: Elementary Theory. Okayasu, R. "A Solution to Haagerup's
Problem on Decomposable Maps." 30 Oct 2026. https://arxiv.org/abs/2610.41746v1 . OpenAI.
"An Isomorphism of the Free Group Factors." Sep. 23, 2026a. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf . OpenAI.
"Relative Generation and the Generator Problem for Finite Factors." Sep. 23,
2026b. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf . OpenAI.
"Universal Strong Kadison-Kastler Stability." Sep. 23, 2026c. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf . Pisier,
G. Tensor Products of C*-Algebras and Operator Spaces: The Connes-Kirchberg Problem.
Cambridge, England: Cambridge University Press, 2020. https://doi.org/10.1017/9781108782081 . Shlyakhtenko,
D. "On the Isomorphism of Semicircular Algebras." 14 Oct 2026. https://arxiv.org/abs/2610.19590 . Takesaki,
M. Theory
of Operator Algebras I. Referenced
on Wolfram|Alpha von Neumann Algebra
Cite this as:
Weisstein, Eric W. , with contributions by Christopher Stover and Mohammad Sal Moslehian . "von
Neumann Algebra." From MathWorld https://mathworld.wolfram.com/vonNeumannAlgebra.html
Subject classifications More... Less...
is a 
-subalgebra
of 
such that 
is equal to its bicommutant 
(Dixmier 1981). Here, 
denotes the algebra of bounded
operators from 
to itself.
-subalgebra of 
is a von Neumann algebra iff it is strongly closed. This
is further equivalent to a number of other analytic properties of 
and of 
(Blackadar 2013), and due to its bijective equivalence
is sometimes used as a definition for von Neumann algebras.
In some literature, the assumption of 
being unital (i.e., 
containing the identity) is added
to the hypotheses of this equivalence though, strictly speaking, the result holds
in the somewhat more general case that 
is merely nondegenerate.
which admits a Banach space 
as a pre-dual. This convention, though
not unheard of, is somewhat rare among literature on the topic.
on 
generators, its group
von Neumann algebra 
is generated by its left regular group
representation.
is the range of a norm-one bounded
projection from 
. Okayasu (2026) proves that
injectivity is equivalent to equality of the decomposable and completely bounded
norms for every linear
map 
, where 
has coordinatewise operations. The decomposable
norm is defined using completely positive maps.
are isomorphic for 
, 3, ... and countably infinite 
. Shlyakhtenko (2026) gives a self-contained adaptation
of the argument and proves the corresponding C*-algebra
result for algebras generated by finite or countably infinite freely
independent semicircular families
of at least two elements.
factor with separable predual is generated by one operator,
equivalently by two self-adjoint operators.
OpenAI (2026c) also announced that sufficiently close unital von Neumann algebras
on a common complex Hilbert space are conjugate
by a unitary arbitrarily close to the identity,
uniformly over the algebras and representations. The
closeness is measured by the Hausdorff distance
between their operator norm unit
balls.
-Algebras and von Neumann Algebras." 2013.