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 $B$?



Approximate units from strictly positive elements in $C^{*}$-algebras.
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.


