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 $B$?
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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


