Implication reversal in convergence under finer/coarser topologies

Question

Just a quick sanity check.

Consider the following three topologies on $B(H)$: WOT, SOT, and $\tau_{\lVert · \rVert}$. As WOT ⊂ SOT ⊂ $\tau_{\lVert · \rVert}$, does it follow, for a subset $A ⊆ B(H)$ and a net $(x_i)_i$, that

$$x_i \xrightarrow{\tau_{\lVert · \rVert}} x \Longrightarrow x_i \xrightarrow{\text{SOT}} x \Longrightarrow x_i \xrightarrow{\text{WOT}} x $$ and $$A = \overline{A}^{WOT} \Longrightarrow A = \overline{A}^{SOT} \Longrightarrow A = \overline{A}^{\tau_{\lVert · \rVert}} ? $$

Answer 1

If we write it

$$ \text{WOT} \subset{\text{SOT}} \subset \tau_{\lVert · \rVert}$$

$$\Longrightarrow \left(x_i \xrightarrow{\tau_{\lVert · \rVert}} x \Longrightarrow x_i \xrightarrow{\text{SOT}} x \Longrightarrow x_i \xrightarrow{\text{WOT}} x \right) $$ $$\Longrightarrow \left( A = \overline{A}^{WOT} \Longrightarrow A = \overline{A}^{SOT} \Longrightarrow A = \overline{A}^{\tau_{\lVert · \rVert}} \right) $$

we could call it a snake lemma when we track "WOT-SOT-norm, WOT-SOT-norm, WOT-SOT-norm" like a mantra and follow it.

                                                                XD