Can we say that $\|e_i a e_i - a\| \to 0\ $?

Question

Let $A$ be a non-unital $C^{\ast}$-algebra. Then for any approximate unit $(e_i)_{i \in I}$ and for any $a \in A$ can we say that $\|e_i a e_i - a e_i \| \xrightarrow{i} 0\ $?

Answer 1

Yes. This is true. Following the usual definition of approximate unit in a $C^*$-algebra (where the approximating elements are contractive), we get $$\|e_i a e_i-ae_i\| =\|(e_i a-a)e_i\| \le \|e_i a-a\|\stackrel{i \to \infty}\longrightarrow 0.$$