C([0,1],M_{l_i}(\mathbb{C}))$. For any $(f,a)\in A$ and $i\in\{1,2,\cdots,l\}$, define $\pi_t:A\rightarrow C([0,1],F_2)$ by $\pi_t(f,a)=f(t)$ and $\pi_t^i:A\rightarrow C([0,1],M_{l_i}(\mathbb{C}))$ by $\pi_t^i(f,a)=f(t,i)$ where $t\in(0,1)$ and $\pi_0^i(f,a)=f(0,i)$ (denoted by $\varphi_0^i(a)$), $\pi_1^i(f,a)=f(1,i)$ (denoted by $\varphi_1^i(a)$). There is a canonical map $\pi_e:\,A\rightarrow F_1$ defined by $\pi_e((f,a))=a$, for all $j=\{1,2,\cdots,p\}$, define $\pi_e^j:\,A\rightarrow M_{k_j}(\mathbb{C})$ by $\pi_e^j((f,a))=a(\theta_j)$.

Define $\widetilde{\pi}_0^j: \varphi_0(F_1)\rightarrow M_{k_j}(\mathbb{C})$ by $$\widetilde{\pi}_0^j\circ\varphi_0(a)={\rm sgn}(\sum_{i=1}^{l}\alpha_{ij})\cdot a(\theta_j)$$ and $\widetilde{\pi}_1^j: \varphi_1(F_1)\rightarrow M_{k_j}(\mathbb{C})$ by $$\widetilde{\pi}_1^j\circ\varphi_1(a)={\rm sgn}(\sum_{i=1}^{l}\beta_{ij})\cdot a(\theta_j)$$ for each $j\in\{1,2,\cdots,p\}$ and ${\rm sgn}(x)$ is the sign function.

In this paper we use the convention that $A=A(F_1,F_2,\varphi_0,\varphi_1),\,B=B(F_1',F_2',\varphi_0',\varphi_1')$, with $$F_1=\bigoplus_{j=1}^p M_{k_j}(\mathbb{C}),\,\,F_2=\bigoplus_{i=1}^lM_{l_i}(\mathbb{C}),\,\,F_1'=\bigoplus_{j'=1}^{p'}M_{k_{j'}'}(\mathbb{C}),\,\,F_2'=\bigoplus_{i'=1}^{l'} M_{l_{i'}'}(\mathbb{C}).$$ Let $\varphi_{0*},\varphi_{1*}$ be represented by matrices $\alpha=({\alpha_{ij}})_{l\times p}$ and $\beta=({\beta_{ij}})_{l\times p}$, and let $\varphi_{0*}',\varphi_{1*}'$ be represented by matrices $\alpha'=({\alpha_{i'j'}'})_{l'\times p'}$ and $\beta=({\beta'_{i'j'}})_{l'\times p'}$.

Let $L(A)=\sum_{i=1}^ll_i$, $L(B)=\sum_{i'=1}^{l'}l_{i'}'$. Denote $\{e_{ss'}^i\}(1\leq i\leq l,\,1\leq s,s'\leq l_i)$ the set of matrix units for $\oplus_{i=1}^p M_{l_i}({\mathbb{C}})$ and $\{f_{ss'}^j\}(1\leq i\leq p,\,1\leq s,s'\leq k_j)$ the set of matrix units for $\oplus_{j=1}^l  M_{k_j}({\mathbb{C}})$.

For each $\eta=\frac{1}{m}$ where $m\in\mathbb{N}_+$. Let $0=x_0<x_1<\cdots<x_{m}=1$ be a partition of $[0,1]$ into $m$ subintervals with equal length $\frac{1}{m}$. We will define a finite subset $H(\eta)\subset A_+$, consisting of two kinds of elements as described below.

(a)For each subset $X_j=\{\theta_j\}\subset Sp(F_1)=\{\theta_1,\theta_2,\cdots,\theta_p\}$ and a list of integers $a_1,b_2,\cdots,a_l,b_l$ with $0\leq a_i<a_i+2\leq b_i\leq m$, denote $W_j\triangleq\cup_{\alpha_{ij}\neq0}[0,a_i]_i\cup\cup_{\beta_{ij}\neq 0}[b_i,1]_i$. Then we call $W_j$ the closed 