bicommutant theorem    The bicommutant theorem characterizes
concrete von Neumann algebras as those concrete C *C^*-algebras
(C *C^*-algebras of bounded operators on some Hilbert space)
that are the commutants of their own commutants.   The
bicommutant theorem (as known as the double commutant theorem,
or von Neumann’s double commutant theorem) is the following
result:   Let AB(H)A \subseteq B(H) be a sub-**-algebra of the
algebra of bounded linear operators on a Hilbert space HH. Then
AA is a von Neumann algebra (and therefore also a C
*C^*-algebra) in HH if and only if A=AA = A'', where AA' denotes
the commutant of AA.  Notice that the condition of AA being a
von Neumann algebra (being closed in the weak operator topology;
“weak” here can be replaced by “strong”, “ultrastrong”, or
“ultraweak” as described in operator topology), which is a
topological condition, is by this result equivalent to an
algebraic condition (being equal to its bicommutant).   Revised
on July 27, 2011 20:40:30 by Toby Bartels (