nLab
relative commutant

Definition

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

$${𝒜}^c(ℬ):=\{B\in ℬ:BA=AB,A\in 𝒜\}$$\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

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