I am reading the proof that, if $I\subset A$ is an ideal in a C$^*$-algebra, then $I$ has an approximate unit $\{e_\lambda\}$ that is quasi-central for $A$, i.e. $e_\lambda a-ae_\lambda\to0$.
The standard proof is due to Arveson and it is presented in Davidson's book, I.9.16. Another presentation of the proof (probably derived from Davidson's book) can be found in this book. However, I am skeptical about a specific part:
The proof starts by taking an approximate unit $\{u_\mu\}_{\mu\in M}$ of $I$ and considering its convex hull, say $N=\text{co}\{u_\mu:\mu\in M\}\subset I$. It is not hard to verify that, under the usual $\leq$ relation induced by positivity in the C$^*$-algebra (and the ideal of course), the set $N$ is directed. Moreover, setting $\nu=f_\nu$ for all $\nu\in N$ we have that $\{f_\nu\}_{\nu\in N}$ is again an approximate unit for $I$.
Now Davidson does the following trick: fix $\mu_0\in M$ and $F=\{a_1,\dots,a_n\}\subset A$ a finite set. Set $N_{\mu\geq\mu_0}:=\text{co}\{u_\mu:\mu\geq\mu_0\}$ (which is again a convex set). Then, by taking a faithful representation $A\subset B(H)$ and representing $A\oplus\dots\oplus A\subset B(H\oplus\dots\oplus H)$ and using Hahn-Banach theorem we prove that there exists $f=f_{F,\mu_0}\in N_{\mu\geq\mu_0}$ such that $$\|(f\oplus\dots\oplus f)(a_1\oplus\dots\oplus a_n)-(a_1\oplus\dots\oplus a_n)(f\oplus\dots\oplus f)\|<\frac{1}{n}$$ hence $\|fa_i-a_if\|<\frac{1}{n}$ for all $i=1,\dots,n$. And then the author concludes the proof by saying that the net $\{f_{F,\mu}\}_{F,\mu}$ (directed by the directed set that is the product of $M$ and the finite subsets of $A$ with the usual inclusion as direction) is a quasicentral approximate unit.
My question is this: I can prove that, yes, indeed $\|af_{F,\mu}-f_{F,\mu}a\|\to0$ for all $a\in A$ and I can also prove that $xf_{F,\mu}\to x$ for all $x\in I$. It is clear to me that $f_{F,\mu}$ are positive contractions. But it is not clear at all why $\{f_{F,\mu}\}_{F,\mu}$ is an increasing net, as approximate units should be. I really cannot understand why it has to be increasing. Or, if it is not necessarily increasing, then how does one "make it" increasing without messing up the other properties?
My idea: it can be observed that $\{f_{F,\mu}\}$ is itself a directed set so I was thinking that maybe I should consider it as a net directed by itself. Then it is trivially increasing, with some tricks I was able to see that it is an approximate unit for $I$ but it stopped being quasicentral: taking $a\in A, \varepsilon>0$, How does one find $f_{F_0,\mu_0}$ so that when $f_{F,\mu}\geq f_{F_0,\mu_0}$ then $\|af_{F,\mu}-f_{F,\mu}a\|<\varepsilon$? when the direction was given by $(F,\mu)$ this was easy: just pick $\mu_0$ arbitrarily and take any finite subset $F_0$ containing $a$ and many other elements so that $1/|F_0|<\varepsilon$. But now this fails.
Any ideas? I feel that the answer is a very simple manipulation that I just cannot see.
Just combine the two orderings: say that $(F,\mu)\preceq(F',\mu')$ if both $(F,\mu)\leq(F',\mu')$ with respect to the product order and $f_{F,\mu}\leq f_{F',\mu'}$. This ordering is still directed: given $(F,\mu)$ and $(F',\mu')$, choose $\nu$ such that $\nu\geq\mu$,$\nu\geq\mu'$, $u_\nu\geq f_{F,\mu}$ and $u_\nu\geq f_{F',\mu'}$, and then $(F\cup F',\nu)\succeq (F,\mu)$ and $(F\cup F',\nu)\succeq(F',\mu')$. With respect to this ordering, the net $(f_{F,\mu})$ is then both increasing and quasicentral.