Bounded operator

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,

$S^{\prime }=\{x\in A:sx=xs\text{for}\text{every}s\in S\}.$

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

Properties

• $S^{\prime }=S^{‴}=S^{\prime \prime \prime \prime \prime }$ - A commutant is its own bicommutant.
• $S^{″}=S^{⁗}=S^{\prime \prime \prime \prime \prime \prime }$ - A bicommutant is its own bicommutant.

See also

• Bicommutant
• von Neumann bicommutant theorem

References

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