When reading the proof the (iii) and (iv), I was confused the statement which is marked red.
According to the condition in (iii), we know that $\chi_{\{||A||_e\}}(A)$ is infinite, but it may not be properly infinite, we cannot use the "halving" lemma to get the decomposition.
1.How to decompose $\chi_{\{||A||_e\}}(A)$ into the sum of two infinite projections?
2.How to decompose $A$ as the direct sum of $A'$ and $\|B\|Q$? If we view $A$ as a matrix, How to prove $A=\begin{bmatrix} A' & & & \\ & & &\|B\|Q \end{bmatrix}$?
If a projection $P$ is infinite, then it has a proper subprojection $P_1$ with $P_1\sim P$. In particular $P_1$ is infinite and it has a proper subprojection $P_2\sim P_1$. Continuing this way you get a chain of proper subprojections $$ P\geq P_1\geq P_2\geq P_3\geq\cdots $$ Let $R=\bigwedge P_j$, and $R_j=P_j-P_{j+1}$. Then $$ P=R+\sum_j R_j=\bigg(R+\sum_{j\ \text{ odd}}R_j\bigg)+\bigg(\sum_{j\ \text{ even}}R_j\bigg) $$ is a sum of two infinite projections.
Knowing that $\chi_{\{||A||_e\}}(A)=P+Q$ with both infinite, we can write $$A\,\chi_{\{||A||_e\}}(A)=\|A\|_e\,\chi_{\{||A||_e\}}(A)=\|A\|_e\,P\oplus \|A\|_eQ.$$ Then $$ A=\overbrace{A\,\chi_{\sigma(A)\setminus\{\|A\|_e\}}(A)+\|A\|_e\,P}^{A'}+\|A\|_e\,Q. $$ And because $P$ is infinite, $\|A'\|_e=\|A\|_e$.
Since the $\{P_n\}$ are decreasing, $\lim_{\rm sot}P_n=R$. Then $$ P-R=R_1+P_1-R=R_1+R_2+P_2-R=\cdots=\sum_{j=1}^nR_j+P_n-R\xrightarrow[\rm sot]{}\sum_{j=1}^\infty R_j. $$