{{\operatorname{id}}})\circ\delta(a^*a)=0$. This implies that for all states $f$ on $C^*(G,P)$ and states $g$ on $C^*(K)$, $$g\circ (f\otimes {{\operatorname{id}}})\circ\delta_\mu(a^*a)=(f\otimes g)\circ\delta_\mu(a^*a)=0.$$

To see that $\delta_\mu(a^*a)=0$, let $\pi_1:C^*(G,P)\to H_1$ and $\pi_2:C^*(K)\to H_2$ be faithful representations. Then $\pi_1\otimes\pi_2$ is a faithful representation of $C^*(G,P)\otimes_{\min}C^*(K)$ on $B(H_1\otimes H_2)$ by [@RaeburnWilliams1998 Corollary B.11]. Fix unit vectors $h\in H_1$, $k\in H_2$. There exists a state $f_h\otimes f_k$ on $C^*(G,P)\otimes_{\min}C^*(K)$ defined by $$f_h\otimes f_k(x)=(\pi_1\otimes\pi_2(x)(h\otimes k)\mid h\otimes k).$$ Since $(f\otimes g)\circ\delta_\mu(a^*a)=0$ for all states $f$ of $C^*(G, P)$ and $g$ of $C^*(K)$, we have $$\begin{aligned}
  0&=f_h\otimes f_k(\delta_\mu(a^*a))\\
  &=(\pi_1\otimes\pi_2(\delta_\mu(a^*a))(h\otimes k)\mid h\otimes k)\\
  &=(\pi_1\otimes\pi_2(\delta_\mu(a))(h\otimes k)\mid \pi_1\otimes\pi_2(\delta_\mu(a)) h\otimes k)\\
  &=\|\pi_1\otimes\pi_2(\delta_\mu(a))(h\otimes k)\|^2.\end{aligned}$$ Hence $\pi_1\otimes\pi_2(\delta_\mu(a^*a))=0$. Since $\pi_1\otimes \pi_2$ is faithful, $\delta_\mu(a^*a))=0$. But $\delta_\mu$ is injective, and hence $a=0$, and $\Psi^\mu$ is faithful.

Next we investigate the structure of $${\operatorname{range}}\Psi^\mu={\overline{{\operatorname{span}}}}\{w_pw_q^*:\mu(p)=\mu(q)\}.$$ In the statement of the next lemma, note that we can view $C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$ as a $C^*$-subalgebra of $C^*(G,P)$.

Let $(G,P)$ and $(K,Q)$ be quasi-lattice ordered groups, and suppose that $\mu:G\to K$ a controlled map. Let $k\in Q$, and let $F$ be a finite subset of $\Sigma_k$. Let $$B_{k,F}:={\operatorname{span}}\{w_\sigma Dw^*_\tau:\sigma,\tau\in F,D\in C^*\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)\}.$$ Then $B_{k,F}$ is a closed $C^*$-subalgebra of ${\overline{{\operatorname{span}}}}\{w_pw_q^*:\mu(p)=\mu(q)\}$.

To see that $B_{k,F}$ is contained in ${\overline{{\operatorname{span}}}}\{w_pw_q^*:\mu(p)=\mu(q)\}$, it suffices to consider $D$ of 