Commutant

In algebra, the commutant of a subset S of a semigroup (such as an algebra or a group) A is the subset S′ of elements of A commuting with every element of S. In other words,

$S'=\{x\in A: sx=xs\ \mbox{for}\ \mbox{every}\ s\in S\}.$

S′ forms a subsemigroup. This generalizes the concept of centralizer in group theory.

Properties

• S' = S''' = S''''' - A commutant is its own bicommutant.
• S'' = S'''' = S'''''' - A bicommutant is its own bicommutant.

See also

• Bicommutant
• von Neumann bicommutant theorem