Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ be a complex Hilbert space, let $\mathcal{S},\mathcal{T}\subseteq\mathcal{L}(H)$, and let $\mathcal{S}'$, $\mathcal{S}''$ and $\mathcal{S}'''$ be the commutant, double commutant and triple commutant. Write $\mathcal{S}^{*}=\{A^{*}:\ A\in\mathcal{S}\}$, and let $\mathbb{N}$ be the set of natural numbers.

1. (Algebra)¶ $I\in\mathcal{S}'$, and $S+T$, $cS$ and $ST$ belong to $\mathcal{S}'$ for all $S,T\in\mathcal{S}'$ and $c\in\mathbb{C}$. If $\mathcal{S}^{*}\subseteq\mathcal{S}$, then $S^{*}\in\mathcal{S}'$ for every $S\in\mathcal{S}'$.

2. (Order)¶ If $\mathcal{S}\subseteq\mathcal{T}$, then $\mathcal{T}'\subseteq\mathcal{S}'$. Moreover $\mathcal{S}\subseteq\mathcal{S}''$ and $\mathcal{S}'''=\mathcal{S}'$.

3. (Weak limits)¶ Let $(T_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{S}'$ and let $T\in\mathcal{L}(H)$ be such that, for all $\xi,\eta\in H$, the sequence $(\langle\eta,T_{k}\xi\rangle)_{k\in\mathbb{N}}$ converges to $\langle\eta,T\xi\rangle$ in $\mathbb{C}$ with the metric $(z,w)\mapsto|z-w|$. Then $T\in\mathcal{S}'$. In particular $T\in\mathcal{S}'$ whenever $T_{k}\to T$ in operator norm.

4. (Conjugation)¶ Let $J$ be a conjugation of $H$, with conjugated maps $JAJ$. For all $A,B\in\mathcal{L}(H)$ and $c\in\mathbb{C}$: $JAJ\in\mathcal{L}(H)$, $\lVert JAJ\rVert_{\mathrm{op}}=\lVert A\rVert_{\mathrm{op}}$, $(JAJ)^{*}=JA^{*}J$, $(JAJ)(JBJ)=J(AB)J$, $J(JAJ)J=A$, $J(A+B)J=JAJ+JBJ$ and $J(cA)J=\overline{c}\,JAJ$. Moreover $(J\mathcal{S}J)'=J\mathcal{S}'J$.