is a closed $C^*$-subalgebra of ${\overline{{\operatorname{span}}}}\{w_pw_q^*:\mu(p)=\mu(q)\}$.

Let $\{\epsilon_x:x\in P\}$ be the usual basis for $\ell^2(P)$. Let $T$ be the covariant representation of $(G,P)$ on $\ell^2(P)$ such that $T_p\epsilon_x=\epsilon_{px}$, and let $\pi_T$ be the corresponding homomorphism of $C^*(G,P)$ onto $C^*_r(G,P)$ such that $\pi_T(w_p)=T_p$. For $k\in Q$ we consider the subspaces $$H_k:=\{\epsilon_{\gamma z}:\gamma\in\Sigma_k,z\in \mu^{-1}(e)\cap P\}.$$

\[isometric on H\_k\] Let $(G,P)$ and $(K,Q)$ be quasi-lattice ordered groups, and suppose that $\mu:G\to K$ is a controlled map. Let $k\in Q$. Then

1.  $H_k$ is invariant for $\pi_T|_{B_{k,F}}$;

2.  \[isometric on H\_k2\] if $\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$ is amenable, then $\pi_T|_{B_{k,F}}$ is isometric on $H_k$.

For (1), let $\sigma,\tau\in F$ and let $x,y\in\mu^{-1}(e)$ and let $\epsilon_{\gamma z}\in H_k$. Then $w_{\sigma}w_{x}w^*_yw^*_{\tau}$ is a spanning element of $B_{k,F}$. Since $\mu(\tau)=k=\mu(\gamma)$ we have $$\pi_T(w_{\sigma}w_{x}w^*_yw^*_{\tau})\epsilon_{\gamma z}=
    \begin{cases}\epsilon_{\sigma xy^{-1} z}&\text{if $\gamma=\tau$ and $y\leq z$}\\0&\text{otherwise.}\end{cases}$$ If $\pi_T(w_{\sigma}w_{x}w^*_yw^*_{\tau})\epsilon_{\gamma z}=0$ we are done. Otherwise, to see that $\epsilon_{\sigma xy^{-1} z}$ is back in $H_k$, suppose that $y\leq z$. Then $y^{-1}z\in P$. Since $\sigma x\in \mu^{-1}(k)\cap P$ we have $\epsilon_{(\sigma x)(y^{-1} z)}\in H_k$. It follows that $H_k$ is invariant for $\pi_T|_{B_{k,F}}$.

For (2) suppose that $\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$ is amenable. We will show that $\pi_T|_{B_{k,F}}$ is faithful on $H_k$. Take $B=\sum_{\sigma,\tau\in F} w_{\sigma}D_{\sigma,\tau}w^*_{\tau}\in B_{k,F}$ and suppose that 