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 $B$?

$\begingroup$ Do you have something to do with BaajJulg picture of KKtheory? $\endgroup$– Kolya IvankovMar 17 '11 at 16:17

$\begingroup$ Sorry, of course you do ^_^ $\endgroup$– Kolya IvankovMar 17 '11 at 16:20

$\begingroup$ It's a small world... $\endgroup$– alterationx10Mar 17 '11 at 16:36

$\begingroup$ Does anyone have a reference for this equivalence? Thanks... $\endgroup$– Sergio A. YuhjtmanNov 5 '12 at 23:41
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 KKTheory", Lemma 1.1.21). So for $b\in B$ and $\epsilon>0$, we can find $c\in B$ with $\bch\<\epsilon$, 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.

$\begingroup$ Thanks a lot, that saves me some time! I had not seen how to effectivily use strict positivity. $\endgroup$ Mar 17 '11 at 15:49

$\begingroup$ Thanks Mattew! It is indeed helpful for my purposes too. $\endgroup$ Mar 17 '11 at 16:16