nLab
commutant

Definition

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

$$B\prime =\{a\in A\mid \forall b\in B:ab=ba\}$$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

$$B\subseteq C\prime \mathrm{iff}C\subseteq B\prime .$$B \subseteq C' \qquad iff \qquad C \subseteq B'.

Hence $B\subseteq B″$ and also $B\prime =B‴$.

Related entries

• bicommutant theorem