Bicommutant

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In algebra, the bicommutant of a subset S of 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 S′′.

The bicommutant of S always contains S. Since S′′′ = (S′′)′ = (S′)′′, it follows that the commutant of the bicommutant of S is equal to the commutant of S. Thus we have:

S ⊆ S′′ = S′′′′ = S′′′′′′ = …,

and

S′ = S′′′ = S′′′′′ = …

See also von Neumann double commutant theorem.