Definition

Let $H$ be an Hilbert Space, $B(H)$ the algebra of bounded operators in $H$ and $\mathcal{F} \subset B(H)$ .

The commutant of $\mathcal{F}$ , usually denoted $\mathcal{F}'$ , is the subset of $B(H)$ consisting of all elements that commute with every element of $\mathcal{F}$ , that is

The double commutant of $\mathcal{F}$ is just $(\mathcal{F}')'$ and is usually denoted $\mathcal{F}''$ .

Properties:

• If $\mathcal{F}_1 \subseteq \mathcal{F}_2$ , then $\mathcal{F}_2' \subseteq \mathcal{F}_1'$ .
• $\mathcal{F} \subseteq \mathcal{F}''$ .
• If $\mathcal{A}$ is a subalgebra of $B(H)$ , then $\mathcal{A} \cap \mathcal{A}'$ is the center of $\mathcal{A}$ .
• If $\mathcal{F}$ is self-adjoint then $\mathcal{F}'$ is self-adjoint.
• $\mathcal{F}'$ is always a subalgebra of $B(H)$ that contains the identity operator and is closed in the weak operator topology.
• If $\mathcal{F}$ is self-adjoint then $\mathcal{F}'$ is a von Neumann algebra.

Remark: The commutant is a particular case of the more general definition of centralizer.