Nuclear dimension of $C^*$-algebras

Question

I met with troubles in the the proof of forward implication. By definition of nuclear dimension of $C^*$-algebras, it is easy to see that (1) and (3) hold, how to prove (2) and (4)?

Answer 1

Fix $x_1,\ldots,x_n\in B_0$ and let $\varepsilon>0$ be given. Since $\dim_{\operatorname{nuc}}(B)\leq d$, there is a finite dimensional $C^*$-algebra $F=F^{(0)}\oplus\cdots\oplus F^{(d)}$ and c.p. maps $\psi:B\to F$, $\varphi:F\to B$ with $\psi$ contractive, $\varphi\mid_{F^{(i)}}$ order zero for $i=0,1,\ldots,d$, and $\|\varphi\circ\psi(x_k)-x_k\|<\varepsilon$ for $k=1,\ldots,n$. Since the nuclear dimension of any finite-dimensional $C^*$-algebra is zero, we have $$\sum_{l=0}^d(\dim_{\operatorname{nuc}}(F^{(l)})+1)(0+1)=d+1.$$