nLab relative commutant

Definition

Let $\mathcal{A} \subset \mathcal{B}$ be an inclusion of $*-$ algebras. The relative commutant $\mathcal{A}^c(\mathcal{B})$ is defined by

$$\mathcal{A}^c(\mathcal{B}) := \{ B \in \mathcal{B} : B A = A B, \; A \in \mathcal{A} \}$$

If the algebras are operator algebras defined on a Hilbert space, then

$$\mathcal{A}^c(\mathcal{B}) = \mathcal{A}' \bigcap \mathcal{B}$$