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