Sot closure of the unit ball of a subalgebra of $B(H)$ Let $A$ be a $C^* -$ subalgebra of B(H) and $S$ be the closed unit ball of $A$. 1- $S$ is convex and bounded, so $ S = weak^* -cl ~S$. (Is it correct?) 2- By the Kapalansky density theorem, we have $sot-cl~ S = (wot-cl~A)_{\|.\|\leq 1}$, and also I know that $sot-cl ~S = w^*-cl~S$. From 1,2, I conclude that $S = (wot-cl A)_{\|.\|\leq 1}$ which is not correct. Where is my mistake. Thanks for your help.

Your 1 is not correct. On a ball, weak $^*=$ wot. Hahn-Banach says that the **weak** and norm closures agree on a bounded convex set.