A non-trivial corollary of the so-called bicommutant theorem says that a nondegenerate-subalgebra of is a von Neumann algebraiff 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.
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.
-Algebras and von Neumann Algebras." 2013.