# commutant is a weak operator closed subalgebra

Let $H$ be a Hilbert space and $B(H)$ the algebra of bounded operators in $H$. Recall that the commutant of a subset $\mathcal{F}\subset B(H)$ is the set of all bounded operators that commute with those of $\mathcal{F}$, i.e.

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

- If $\mathcal{F}\subset B(H)$, then $\mathcal{F}^{\prime}$ is a subalgebra of $B(H)$ that contains the identity operator and is closed in the weak operator topology.

: It is clear that $\mathcal{F}^{\prime}$ contains the identity operator, since it commutes with all operators in $B(H)$ and in particular with those of $\mathcal{F}$.

Let us now see that $\mathcal{F}^{\prime}$ is a subalgebra of $B(H)$. Let $T_{1},T_{2}\in\mathcal{F}^{\prime}$ and $\lambda\in\mathbb{C}$. We have that, for all $S\in\mathcal{F}$,

 $\displaystyle S(T_{1}+T_{2})=ST_{1}+ST_{2}=T_{1}S+T_{2}S=(T_{1}+T_{2})S$ $\displaystyle S(\lambda T_{1})=\lambda ST_{1}=\lambda T_{1}S$ $\displaystyle S(T_{1}T_{2})=T_{1}ST_{2}=T_{1}T_{2}S$

thus, $T_{1}+T_{2}$, $\lambda T_{1}$ and $T_{1}T_{2}$ all belong to $\mathcal{F}^{\prime}$, and therefore $\mathcal{F}^{\prime}$ is a subalgebra of $B(H)$.

It remains to see that $\mathcal{F}^{\prime}$ is weak operator closed. Suppose $(T_{i})$ is a net in $\mathcal{F}^{\prime}$ that converges to $T$ in the weak operator topology. Then, for all $x,y\in H$ we have that $\langle T_{i}x,y\rangle\to\langle Tx,y\rangle$. Thus, for all $S\in\mathcal{F}$, we have

 $\displaystyle\langle(TS-ST)x,y\rangle$ $\displaystyle=$ $\displaystyle\langle TSx,y\rangle-\langle Tx,S^{*}y\rangle$ $\displaystyle=$ $\displaystyle\lim\big{(}\langle T_{i}Sx,y\rangle-\langle T_{i}x,S^{*}y\rangle% \big{)}$ $\displaystyle=$ $\displaystyle\lim\,\langle(T_{i}S-ST_{i})x,y\rangle$ $\displaystyle=$ $\displaystyle\lim\,\langle(T_{i}S-T_{i}S)x,y\rangle$ $\displaystyle=$ $\displaystyle 0$

Hence, $TS-ST=0$, so that $T\in\mathcal{F}^{\prime}$. We conclude that $\mathcal{F}^{\prime}$ is closed in the weak operator topology. $\square$

Title commutant is a weak operator closed subalgebra CommutantIsAWeakOperatorClosedSubalgebra 2013-03-22 18:39:32 2013-03-22 18:39:32 asteroid (17536) asteroid (17536) 7 asteroid (17536) Theorem msc 46L10