Bicommutant
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In algebra, the bicommutant of a subset S of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written 
.
The bicommutant of S always contains S. So 
. On the other hand, 
. So 
, i.e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:
and
for n > 1.
It is clear that, if S1 abd S2 are subsets of a semigroup,
If it is assumed that 
and 
(this is the case, for instance, for von Neumann algebras), then the above equality gives
