that $$\phi(f,a)=u^*\cdot{\rm diag}\big(\underbrace{a(\theta_1),
 \cdots,a(\theta_1)}_{t_1},\cdots,\underbrace{a(\theta_p),\cdots,a(\theta_p)}_{t_p},f(y_1),\cdots,f(y_{\bullet})\big)\cdot u,$$ where $y_1,y_2,\cdots,y_{\bullet}\in \coprod_{i=1}^{l}[0,1]_i$. We shall use $\bullet$ to denote any possible positive integer. For $y=(0,i)$(also denoted by $0_i$), one can replace $f(y)$ by $$\big(\underbrace{a(\theta_1),
 \cdots,a(\theta_1)}_{\alpha_{i1}},\cdots,\underbrace{a(\theta_p),\cdots,a(\theta_p)}_{\alpha_{ip}}\big)$$ in the above expression, and do the same with $y=(1,i)$. After this procedure, we can assume each $y_k$ is strictly in the open interval $(0,1)_i$ for some $i$. We write the spectrum of $\phi$ by $$Sp\phi=\{\theta_1^{\thicksim t_1},\theta_2^{\thicksim t_2},\cdots,\theta_p^{\thicksim t_p},y_1,y_2,\cdots,y_{\bullet}\},$$ where $y_k\in \coprod_{i=1}^{l}(0,1)_i$ and we use $\{{\theta_j}^{\thicksim t_j}\}$ to denote $\{\underbrace{\theta_j,\cdots,\theta_j}_{t_j\ times}\}$.

Let $\omega_i=\#(Sp\phi\cap(0,1)_i)$ denote the number of $y_k$’s which are in the $i^{th}$ open interval $(0,1)_i$ counting multiplicity. If $\phi$ is unital then we have $$\sum_{j=1}^{p}t_jk_j+\sum_{i=1}^{l}\omega_il_i=n.$$

If $f=f^*\in A$, we use $Eg(\phi(f))$ to denote the eigenvalue list of $\phi(f)$, then

$$\#(Eg(\phi(f)))=n\,\,{\rm(counting\,\,multiplicity)}.$$

[ Let us use $\theta_1,\theta_2,\cdots,\theta_p$ denote the spectrum of $F_1$ and denote the spectrum of $C([0,1],F_2)$ by $(t,i)$, where $0\leq t\leq1$ and $i\in \{1,2,\cdots,l\}$ indicates that it is in $i$th block of $F_2$. So $$Sp(C([0,1],F_2))=\coprod_{i=1}^{l}\{(t,i),\,0\leq t\leq1\}.$$ Using identification of $f(0)=\varphi_0(a)$ and $f(1)=\varphi_1(a)$ for $(f,a)\in A,\,(0,i)\in Sp(C[0,1])$ is identified with $$(\theta_1^{\thicksim\alpha_{i1}},\theta_2^{\thicksim\alpha_{i2}},\cdots,\theta_p^{\thicksim\alpha_{ip}})\subset Sp(F_1)$$ and $(1,i)\in Sp(C([0,1],F_2))$ is identified with $$(\theta_1^{\thicksim\beta_{i1}},\theta_2^{\thicksim\beta_{i2}},\cdots,\theta_p^{\thicksim\beta_{ip}})\subset Sp(F_1)$$ as in $Sp(A)=Sp(F_1)\cup \coprod_{i=1}^{l}(0,1)_i$. ]{}

Let $A=A(F_1,F_2,\varphi_0,\varphi_1)$. Written $a\in F_1$ as $a=(a(\theta_1),a(\theta_2),\cdots,a(\theta_p))$, $f(t)\in C([0,1],F_2)$ as $$f(t)=(f(t,1),f(t,2),\cdots,f(t,l))$$ where $a(\theta_j)\in M_{k_j}(\mathbb{C})$, $f(t,i)\in 