Given a Hilbert space, a -subalgebra of is said to be a von Neumann algebra in provided that is equal to its bicommutant (Dixmier 1981). Here, denotes the algebra of bounded
operators from
to itself.
A non-trivial corollary of the so-called bicommutant theorem says that a nondegenerate-subalgebra of is a von Neumann algebra if and only if 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.
One can easily show that every von Neumann algebra is a W-*-algebra and contrarily; as a result, some literature defines a von Neumann algebra as a C-*-algebra which admits a Banach space as a pre-dual. This convention, though
not unheard of, is somewhat rare among literature on the topic.
, a 
-subalgebra 
of 
is said to be a von Neumann algebra in 
provided 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 if and only if 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.
-Algebras and von Neumann Algebras." 2013.