The hereditary subalgebra

Question

If $B$ is a C*-algebra and $A\subset B$ is a hereditary subalgebra, then , taking $\{e_{n}\}$ be the approximate unit of $A$, can we verify $e_{n}be_{n} \in A$ for every $b\in B$?

Answer 1

The following theorem can be found in Murphy's C*-algebra and operator theory book. I changed the letters in the theorem according to your question.

Theorem. Let $A$ be a C*-subalgebra for a C*-algebra $B$. Then $A$ is hereditary in $B$ if and only if $aba'\in A$ for all $a,a'\in A$ and $b\in B$.