Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The existence of a countable approximate unit in a $C^{*}$-algebra $B$ is equivalent to the existence of a strictly positive element $h\in B$. There are several ways to construct an approximate unit from $h$. My question is, does $h(h+\frac{1}{n})^{-1}$ constitute an approximate unit for $B$?

share|improve this question
Do you have something to do with Baaj-Julg picture of KK-theory? –  Kolya Ivankov Mar 17 '11 at 16:17
Sorry, of course you do ^_^ –  Kolya Ivankov Mar 17 '11 at 16:20
It's a small world... –  alterationx10 Mar 17 '11 at 16:36
Does anyone have a reference for this equivalence? Thanks... –  Sergio A. Yuhjtman Nov 5 '12 at 23:41

1 Answer 1

up vote 5 down vote accepted

I think this works: Functional calculus shows that $h h (h+1/n)^{-1} \rightarrow h$. Then $h$ is strictly positive if and only if $Bh$ is dense in $B$ (See, for example, Jensen+Thomsen, "Elements of KK-Theory", Lemma 1.1.21). So for $b\in B$ and $\epsilon>0$, we can find $c\in B$ with $\|b-ch\|<\epsilon$, and for all $n$ sufficiently large, also $\|ch h(h+1/n)^{-1} - ch\| < \epsilon$. Thus \begin{align*} &\| b - bh(h+1/n)^{-1} \| \\&< \epsilon + \| ch - chh(h+1/n)^{-1}\| + \|chh(h+1/n)^{-1} - bh(h+1/n)^{-1}\| \\ &< 2\epsilon + \epsilon \|h(h+1/n)^{-1}\| < 3\epsilon. \end{align*} Thus we're done.

share|improve this answer
Thanks a lot, that saves me some time! I had not seen how to effectivily use strict positivity. –  alterationx10 Mar 17 '11 at 15:49
Thanks Mattew! It is indeed helpful for my purposes too. –  Kolya Ivankov Mar 17 '11 at 16:16

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.