I am reading a book "C*-algebra and finite-Dimensional Approximations". I can not understand the proof of the excision theorem in the fundamental facts of the book.
Theorem 1.4.10(Excision) Let $A$ be a C*-algebra and $\phi$ be a pure state. There exists a net $(e_{i})\subset A$ such that $0\leq e_{i}\leq1$, $\phi(e_{i})=1$ and $\lim_{i}||e_{i}ae_{i}-\phi(a)e_{i}^{2}||=0$ for every $a\in A$.
Proof. We first assume that $A$ is unital. Let $L=\{a \in A: \phi(a^{\ast}a)=0\}$ be the left ideal associated to $\phi$ and $\{c_{i}\}$ be a right approximate unit for $L$( i.e., $\{c_{i}\}$ is an approximate unit for the hereditary subalgebra $L\cap L^{\ast}$ and for every $a\in L$ we have $||a-ac_{i}||\rightarrow0$). Let $e_{i}=1-c_{i}$. Since $a-\phi(a)\in \ker\phi=L+L^{\ast}$(if $\phi$ is a pure state, this equation is always true), we have $\lim||e_{i}(a-\phi(a))e_{i}||=0$ and $\phi(e_{i})=1$.
Now suppose that $A$ is nonunital and take $e_{i}$ as above for $\tilde{A}$(the unitization of A). Let $\{b_{j}\}$ be a quasicentral approximate unit for $A$ such that $\phi(b_{j})=1$. (Existence follows from Kadison's Transitinity Theorem.) Then, $b_{j}e_{j}b_{j}$ does the job.
My question: I could not understand the sencond part of the proof. Could someone explain to me and show me some details? Thanks.
You have that $A$ is a maximal ideal in its unitization. Then you can construct in $\tilde A$ a quasicentral approximate unit $\{b_j'\}\subset A$. Note also that a state on $A$ has a unique extension to a state in $\tilde A$; let us still call this extension $\phi$.
We also have $\limsup \phi(b_j')=1$: if $\phi(b_j')<1-\delta$ for all $j$, then $1=\phi(I)=\lim\phi(b_j'I)<1-\delta$. So by choosing a subset if necessarily, we can assume that $\phi(b_j')>1/2$, say, and so $b_j=b_j'/\phi(b_j')$ gives us our quasicentral approximate unit with $\phi(b_j)=1$.
Now apply the first part of the proof to the unital algebra $\tilde A$, to get $\{e_i\}$. Consider the net $\{b_je_kb_j\}_{k,j}$ (two indices, I don't personally see how to make the proof work if $k=j$). It is straightforward to check that this is again an approximate unit in $\tilde A$. So arguing as before we can renormalize and choose a subset to get the condition $\phi(b_je_kb_j)=1$. Fix $\varepsilon>0$; for any $a\in A$, for $k$ big enough we have $\|e_kae_k-\phi(a)e_k^2\|<\varepsilon$, and (recall that $\{b_j\}$ is quasicentral) $$ \limsup_j\|b_je_kb_j\,a\,b_je_kb_j-\phi(a)(b_je_kb_j)^2\|=\limsup_j\|b_j^2(e_kae_k-\phi(a)e_k^2)b_j^2\|<\|e_kae_k-\phi(a)e_k^2\|<\varepsilon. $$