If $A$ is a C*-algebra, you can use Kaplansky denisity theorem like this: Suppose $\newcommand{\cl}{\operatorname{cl_{SOT}}}$$a\in \cl A$,then $a/\|a\|\in (\cl A)_1=\cl (A_1)=A_1$, hence $a\in A$.
C. Ding
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