commutant

Definition

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

The commutant of $ℱ$, usually denoted ${ℱ}^{\prime }$, is the subset of $B(H)$ consisting of allelements that commute with every element of $ℱ$, that is

${ℱ}^{\prime }=\{T\in B(H):TS=ST,\forall S\in ℱ\}$

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

Properties:

• If ${ℱ}_1\subseteq {ℱ}_2$, then ${ℱ}_2^{\prime }\subseteq {ℱ}_1^{\prime }$.

• $ℱ\subseteq {ℱ}^{\prime \prime }$.

• If $𝒜$ is a subalgebra of $B(H)$, then $𝒜\cap {𝒜}^{\prime }$ is the center (http://planetmath.org/CenterOfARing) of $𝒜$.

• If $ℱ$ is self-adjoint then ${ℱ}^{\prime }$ is self-adjoint.

• ${ℱ}^{\prime }$ is always a subalgebra of $B(H)$ that contains the identity operator and is closed in the weak operator topology.

• If $ℱ$ is self-adjoint then ${ℱ}^{\prime }$ is a von Neumann algebra.

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