If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

Question

The following is a remark of Murphy's C*-algebras and operator theory:

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I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is a C*-algebra)so $ab^{1/2}\in I$ and $J$ is an ideal of $I$, thus $ab=(ab^{1/2})b^{1/2}\in J$. Where is my mistake? Please help me. Thanks in advance.

Answer 1

I think your argument is fine. I fail to see why Murphy feels the need to use approximate units in this argument.