Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
up vote 1 down vote accepted

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.