Averson's 1970 paper on extensions of $C^*$-algebras seems to assume that every ideal has an approximate identity.
However, I am a little bit suspicious here, since he does not assume the closeness of these ideals-at certain steps, he proves something for the ideal of all finite rank operators, which is not closed.
Since closed ideals of a $C^*$-algebra are themselves $C^*$-algebras, we know that closed ideals have approximate identities. However, if we leave out the condition of closeness, how can we show that an ideal still has an approximate identity? Or, in other words, when you remove the 'skin' of an closed ideal, how can you be sure that enough elements in the approximate identity remains there?