$$(\delta\otimes {{\operatorname{id}}})\circ\delta=({{\operatorname{id}}}\otimes \delta_G)\circ \delta.$$ We say that $\delta$ is *nondegenerate* if $\delta(A)(1\otimes C^*(G))=A\otimes_{\min} C^*(G)$.

\[faithful coaction exists\] Let $(G,P)$ be a quasi-lattice ordered group. Suppose that there exists a group $K$ and a homomorphism $\mu:G\to K$. Then there exists an injective coaction $$\delta_\mu:C^*(G,P)\to C^*(G,P)\otimes_{\min} C^*(K)$$ characterised by $\delta_\mu(w_p)=w_p\otimes u_{\mu(p)}$ for all $p\in P$.

Let $W:P\to C^*(G,P)\otimes_{\min}C^*(K)$ be characterised by $W_p=w_p\otimes u_{\mu(p)}$. We will show that $W$ is a covariant representation, and then take $\delta_\mu:=\pi_W$. Unitaries are isometries and hence $W_p$ is isometric for all $p\in P$. Observe that $W_e=w_e\otimes u_{\mu(e)}=1\otimes 1$, and $$W_pW_q=w_pw_q\otimes u_{\mu(p)}u_{\mu(q)}=w_{pq}\otimes u_{\mu(pq)}=W_{pq}\text{ for all $p,q\in P$.}$$ Thus $W$ is an isometric representation. To prove $W$ is covariant, we fix $x,y\in P$ and compute: $$\begin{aligned}
      W_xW^*_xW_yW^*_y&=w_xw^*_xw_yw^*_y\otimes u_{\mu(x)}u^*_{\mu(x)}u_{\mu(y)}u^*_{\mu(y)}\\
      &=\begin{cases}w_{x\vee y}w^*_{x\vee y}\otimes 1&\text{if $x\vee y<\infty$}\\0\otimes 1&\text{otherwise}\end{cases}\\
      &=\begin{cases}w_{x\vee y}w^*_{x\vee y}\otimes w_{\mu(x\vee y)}w^*_{\mu(x\vee y)}&\text{if $x\vee y<\infty$}\\0&\text{otherwise}\end{cases}\\
      &=W_{x\vee y}W^*_{x\vee y}.
    \end{aligned}$$ Thus $W$ is a covariant representation of $P$. By the universal property of $C^*(G,P)$, there exists a homomorphism $\delta_\mu:=\pi_W$, which has the desired properties. Since $W_e=1\otimes 1$ it follows that $\delta_\mu$ is unital.

To prove the comultiplication identity, we compute on generators: for $p,q\in P$ we have $$\begin{aligned}
      ((\delta_\mu\otimes{{\operatorname{id}}})\circ \delta_\mu)(w_pw^*_q)&=(\delta_\mu\otimes{{\operatorname{id}}})(w_pw^*_q\otimes u_{\mu(pq^{-1})})\\
      &=\delta_\mu(w_pw^*_q)\otimes {{\operatorname{id}}}(u_{\mu(pq^{-1})})\\
      &=w_pw^*_q\otimes u_{\mu(pq^{-1})}\otimes u_{\mu(pq^{-1})}\\
      &=w_pw^*_q\otimes \delta_K(u_{\mu(pq^{-1})})\\
      &={{\operatorname{id}}}\otimes\delta_K(w_pw^*_q\otimes u_{\mu(pq^{-1})})\\
      &=(({{\operatorname{id}}}\otimes\delta_K)\circ \delta_\mu)(w_pw^*_q).
    