nLab commutant

Definition

In an associative algebra $A$, the commutant of a set $B \subset A$ of elements of $A$ is the set

$$B' = \{a \in A | \forall b \in B: a b = b a \}$$

of elements in $A$ that commute with all elements in $B$.

Properties

The operation of taking a commutant is a contravariant map $P(A) \to P(A)$ that is adjoint to itself in the sense of Galois connections. In other words, we have for any two subsets $B, C \subseteq A$ the equivalence

$$B \subseteq C' \qquad iff \qquad C \subseteq B'.$$

Hence $B \subseteq B''$ and also $B' = B'''$.

Related entries

• bicommutant theorem

• centralizer