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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?


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up vote 1 down vote accepted

You can find the proof of this theorem at page 7 in this lecture notes.

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Thanks, Norbert! That's a good proof. BTW, also on page7 the author mentioned that Connes advocates viewing elements of these ideals as infinitesimals, do you have an idea what does that mean? Thanks! – Hui Yu Jul 12 '12 at 1:33
Informally compact operators are "small" operators. And one may want to study operators up to compact perturbations. Since compact operators forms closed ideal you can factorize $\mathcal{B}(H)$. In the first case you will get Calkin algebra used in study of Fredholm operators. – Norbert Jul 12 '12 at 10:59

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