$(G,P)$ and $(K,Q)$ be quasi-lattice ordered groups, and suppose that $\mu:G\to K$ is a controlled map. Then $\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$ is a quasi-lattice ordered group.

It is clear that $\mu^{-1}(e)$ is a subgroup of $G$ and that$\mu^{-1}(e)\cap P$ is a unital semigroup. Suppose that $x,y\in \mu^{-1}(e)$ have a common upper bound $z\in \mu^{-1}(e)\cap P$. We know that $z$ is a common upper bound for $x,y$ in $P$, and hence $x\vee y$ exists in $P$ and $x\vee y\leq z$. Now $\mu(x\vee y)=\mu(x)\vee \mu(y)=e$, and hence $x\vee y\in \mu^{-1}(e)\cap P$. Thus $\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$ is a quasi-lattice ordered group.

To prove Theorem \[p-controlled map implies amenable\] we will show that the conditional expectation $$E:C^*(G,P)\to \overline{{\operatorname{span}}}\{w_pw^*_p:p\in P\}$$ is faithful. We will use the amenability of $K$ to construct a faithful conditional expectation $\Psi^\mu:C^*(G,P)\to{\overline{{\operatorname{span}}}}\{w_pw^*_q:\mu(p)=\mu(q)\}$, and then show that $E$ is faithful when restricted to ${\operatorname{range}}\Psi^\mu$. To construct $\Psi^\mu$ we follow the method of [@LacaRaeburn1996 Lemma 6.5] which uses a coaction.

Let $G$ be a discrete group and let $A$ be a unital $C^*$-algebra. Let $$\delta_G:C^*(G)\to C^*(G)\otimes_{\min} C^*(G)$$ be the comultiplication of $G$ which is characterised by $\delta_G(u_g)=u_g\otimes u_g$ for $g\in G$. A *coaction* of $G$ on $A$ is a unital homomorphism $\delta:A\to A\otimes_{\min} C^*(G)$ such that 