the form $w_\alpha w_\beta^*$ where $\mu(\alpha)=\mu(\beta)=e$. Then $w_\sigma Dw^*_\tau=w_{\sigma\alpha}w_{\tau\beta}^*$ and $\mu(\sigma\alpha)=\mu(\sigma)=k=\mu(\tau)=\mu(\tau\beta)$. Thus $B_{k,F}$ is a subalgebra of ${\overline{{\operatorname{span}}}}\{w_pw_q^*:\mu(p)=\mu(q)\}$.

We will prove the lemma by showing that $B_{k,F}$ is isomorphic to $$M_{F}({\mathbb{C}})\otimes C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big).$$ By Definition \[expanded controlled map definition\], the elements of $F$ have no common upper bound unless they are equal. So $$(w_{\sigma}w^*_{\tau})(w_{\gamma}w^*_{\delta})=w_{\sigma\tau^{-1}(\tau\vee\gamma)}w^*_{\delta\gamma^{-1}(\tau\vee\gamma)}=\begin{cases}w_{\sigma}w^*_{\delta}&\text{if $\tau=\gamma$}\\0&\text{otherwise.}\end{cases}$$ Thus $\{w_{\sigma}w^*_{\tau}:\sigma,\tau\in F\}$ is a set of matrix units in the $C^*$-algebra $\overline{B_{k,F}}$. This gives a homomorphism $\theta:M_{F}({\mathbb{C}})\to \overline{B_{k,F}}$ which maps the matrix units $\{E_{\sigma,\tau}:\sigma,\tau\in F\}$ in $M_{F}({\mathbb{C}})$ to $\{w_{\sigma}w^*_{\tau}:\sigma,\tau\in F\}\subset {B_{k,F}}$. It is easy to check that the formula $$\psi(D)=\sum_{\gamma\in F}w_{\gamma}Dw^*_{\gamma}$$ gives a homomorphism $\psi:C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)\to B_{k,F}$. We have $$\begin{aligned}
      \theta(E_{\sigma,\tau})\psi(D)&=w_{\sigma}w^*_{\tau}\sum_{\gamma\in F}w_{\gamma}Dw^*_{\gamma}\\
      &=w_{\sigma}w^*_{\tau}w_{\tau}Dw^*_{\tau}\quad\text{($w^*_{\tau}w_{\gamma}=0$ unless $\tau=\gamma$)}\\
      &=w_{\sigma}Dw^*_{\tau}\\
      &=(\sum_{\gamma\in F}w_{\gamma}Dw^*_{\gamma})w_{\sigma}w^*_{\tau}\\
      &=\psi(D)\theta(E_{\sigma,\tau}).
    \end{aligned}$$ Each $A\in M_{F}({\mathbb{C}})$ is a linear combination of the $E_{\sigma,\tau}$, and hence $\psi(D)\theta(A)=\theta(A)\psi(D)$ for all $A\in M_{F}({\mathbb{C}})$ and $D\in C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$. Since the ranges of $\theta$ and $\psi$ commute, the universal property of the maximal tensor product gives a homomorphism $\theta\otimes_{\max} \psi$ of $M_{F}({\mathbb{C}})\otimes_{\max}C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$ into $\overline{B_{k,F}}$.

By [@RaeburnWilliams1998 Theorem B.18] $$\begin{aligned}
M_{F}({\mathbb{C}})\otimes&_{\max}C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)\\
&={\operatorname{span}}\{E_{\sigma,\tau}\otimes D: \sigma,\tau\in F \text{\ and\ } D\in C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big) \},\end{aligned}$$ with no closure. So the range of $\theta\otimes_{\max} \psi$ is spanned by $\theta(E_{\sigma,\tau})\psi(D)=w_{\sigma}Dw^*_{\tau}$ and hence is $B_{k,F}$. Thus $B_{k,F}$ 