How to show strong operator continuity for zero implies in ball

Question

Let $\eta: M \mapsto B(\mathcal{K})$ is strong operator continuous at zero, we are putting vN algebra $M$ with Strong operator topology and $B(\mathcal{K})$ Weak operator topology, how to say $\eta$ is continuous on $(M)_2$, the ball of radius 2??

Answer 1

I'm not sure if this is what you are asking: if $x_j\to x$, then $x_j-x\to0$ (and conversely). So if $\eta $ is continuous at zero and linear, $$ \lim_j\eta(x_j)=\lim_j \eta(x_j-x)+\eta(x)=\eta(x). $$ That continuity at zero implies continuity everywhere holds for any linear topology and any linear function (with the above argument).