Bicommutant
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In algebra, the bicommutant of a subset <math>S<math> 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 <math>S''<math>.
The bicommutant of <math>S<math> always contains <math>S<math>. Since <math>S'''=(S'')' = (S')''<math>, it follows that the commutant of the bicommutant of <math>S<math> is equal to the commutant of <math>S<math>. Thus we have
<math>S\subseteq S'' = S'''' = S''''''=\dots<math>,
<math>S' = S''' = S''''' = \dots<math>.
See also von Neumann double commutant theorem.