$B$, this problem can be solved by using the same technique as in case 2 in step 2, we can obtain a projection new$p_j\in B$.

Set $$P_j(x)=\sum_{s=1}^{k_j}\overline{f}_{s1}^j(x)\cdot{\rm new}\,p_j(x)\cdot\overline{f}_{1s}^j(x).$$ Now we have $P_j(x)\leq E_{\widetilde{W_j}}(x)$, and $P_{j_1}(x)P_{j_2}(x)=0$, if $j_1\neq j_2$.

Since $$\phi'(g)P_j=P_j\phi'(g),$$ then $P_j(x)$ almost commutes with $\phi(g)(x)$, $$\begin{aligned}
        \|P_j\phi(g)-\phi(g)P_j\| &=& \|P_j\phi(g)-P_j\phi'(g)+\phi'(g)P_j-\phi(g)P_j\| \\
      &\leq & \|P_j\phi(g)-P_j\phi'(g)\|+\|\phi'(g)P_j-\phi(g)P_j\|  \\
      &\leq & \frac{2}{J},
   \end{aligned}$$ for all $g\in G$.

From the two cases above, we show that for each $\widetilde{V_j}$, we can find a projection $P_j$ almost commutes with $\phi(g)$, for all $g\in G$ and ${\rm rank}\,\,P_j=k_j\#\Lambda_1(h_j)$ for all $j\in \{1,2,\cdots,p\}$.

$Step\,4$: From setp 2 and step 3, we have constructed a collection of mutually orthogonal projections $$P_1^1,\cdots,P_{K-1}^1,\cdots,P_1^l,\cdots,P_{K-1}^l,P_1,P_2,\cdots,P_p.$$ Each of them almost commutes with $\phi(g)$ for all $g\in G$.

Define $$q=1-\sum_{j=1}^{p}P_j-\sum_{i=1}^{l}\sum_{r=1}^{K-1}P_r^i.$$

Change all the spectra in $Sp(\phi_x')\cap W_r^i$ (equal to $Sp(\phi_x)\cap W_r^i$) to $z_r^i$ for each $i=1,2,\cdots,l$, $r=1,2,\cdots,K-1$. We will obtain a pointwise homomorphism $\phi''$ from $\phi'$.

Define $\psi: A\,\rightarrow (1-q)B(1-q)$ by $$\psi(f)=(1-q)\phi''(f)(1-q).$$ Then $\psi$ can be factored through a finite dimensional $C^*$-algebra. Now we have $$\begin{aligned}
        \|\phi(g)q-q\phi(g)\|  &=& \|\phi(g)q-q\phi(g)q+q\phi(g)q-q\phi(g)\| \\
      &\leq& \|(1-q)\phi(g)q\|+\|q\phi(g)(1-q)\| \\
      &=& \|\big((1-q)\phi(g)-\phi(g)(1-q)\big)q\|+\|q\big((1-q)\phi(g)-\phi(g)(1-q)\big)\| \\
      &<& \frac{2}{J}+\frac{2}{J}=\frac{4}{J},
   \end{aligned}$$ and $$\|\phi(g)-q\phi(g)q\oplus\psi(g)\|<\frac{4}{J},$$ for any $g\in G$. This means that conditions (2) and (3) hold.

It is 