I am reading the proof of homotopy invariance of quasi-diagonality from Brown and Ozawa (proposition 7.3.5). There is a technical part in the proof that I cannot understand. The proof is long, so I will try to include only the necessary parts.
So suppose that we have a $C^*$-algebra $B\subset B(H)$ and a finite subset $F\subset B$ and $\varepsilon>0$ is a positive quantity. It is (indeed) not hard to find a finite rank projection $q\in B(H)$ so that $$\|qbq\|\geq\|b\|-\varepsilon $$ for all $b\in F$.
Now assume that, for some integer $n$ (that we shouldn't really worry about) we can find finite rank operators $F_0,\dots, F_n$ so that $$q\leq F_0\leq F_1\leq F_n\leq 1_H,$$ $F_iF_{i+1}=F_i$ for all $i$ and the last one, namely $F_n$, is a projection.
We now set $G_0=F_0$, $G_1=F_1-F_0,\dots, G_n=F_n-F_{n-1}$ (which are all positive, finite rank operators) and we define an operator $$V:H\to H^{n+1},\;\;\;V\xi=(G_0^{1/2}\xi,\dots,G_n^{1/2}\xi).$$ It is not hard to verify that $V^*V=F_n$, so $V$ is a partial isometry and thus $VV^*\in B(H^{n+1})$ is a projection, we set $P=VV^*$.
Now we have a representation $\pi:B\to B(H^{n+1})$ as diagonal operators, namely $\pi(b)=(b,\rho_1(b),\dots,\rho_n(b))$ where $\rho_i:B\to B(H)$ are $*$-homomorphisms that are irrelevant to this question, I think.
The part that I do not understand is this: the authors say: "Since $F_0\geq q$, it is not hard to see that $$\|P\pi(b)P\|\geq\|qbq\|\geq\|b\|-\varepsilon$$ for all $b\in F$." Why is the first inequality so obvious? I cannot prove it.
Doing some calculations one verifies that the matrix of $P$ is equal to $[G_i^{1/2}G_j^{1/2}]_{i,j=0}^n$ and with some calculations one sees that if $j>i+1$ then $G_i^{1/2}G_j^{1/2}=0$ (this follows from the fact that $F_iF_{i+1}=F_i$). So the matrix of $P$ is tridiagonal. I do not know if this helps, I was not able to use this in anyway.
Any ideas on what is implied here? I also looked up Voiculescu's paper and he also gives no explanation of this inequality, so I guess this follows directly from something that I cannot see.
The observation that is not mentioned in the book is this: since $q\leq F_j\leq 1$, $$ \langle q\xi,q\xi\rangle=\langle q\,q\xi,q\xi\rangle\leq\langle F_j\,q\xi,q\xi\rangle\leq\langle q\xi,q\xi\rangle. $$ So $q(F_j-q)q=0$. Then $q(1-F_j)q=0$. As $1-F_j\geq0$, it follows that $(1-F_j)q=0$, and so $qF_j=q$. This implies that $qG_0=q$, $qG_j=0$ for all $j\geq1$.
Then \begin{align} \|P\pi(b)P\| &=\|V^*\pi(b)V\|\\[0.3cm]&\geq\|qV^*\pi(b)Vq\| =\Big\|\sum_jqG_j^{1/2}\rho_j(b)G_j^{1/2}q\Big\|\\[0.3cm] &=\|qG_0^{1/2}bG_0^{1/2}q\|=\|G_0^{1/2}qbqG_0^{1/2}\| =\|G_0^{1/2}qb^*qG_0qbqG_0^{1/2}\|^{1/2}\\[0.3cm] &=\|G_0^{1/2}qb^*qbqG_0^{1/2}\|^{1/2} =\|qbqG_0qb^*q\|^{1/2}\\[0.3cm] &=\|qbqb^*q\|^{1/2}=\|qbq\|. \end{align}