diagonal map $\Delta:B(\ell^2(P))\to B(\ell^2(P))$ given by $$\Delta(T)=\sum_{z\in P}P_zTP_z$$ is a conditional expectation such that $\Delta\circ \pi_T=\pi_T\circ E$ and is faithful.

Now suppose that $R\in C^*(G,P)$ and $E(R^*R)=0$. Then $E(\Psi^{\mu}(R^*R))=0$ and so $\pi_T\circ E(\Psi^{\mu}(R^*R))=0$. This gives $\Delta\circ \pi_T(\Psi^\mu(R^*R))=0$. Since $\Delta$ is faithful, it follows that $\pi_T(\Psi^\mu(R^*R))=0$. Since $\big(\mu^{-1}(e),\mu^{-1}(e)\cap P\big)$ is amenable, Lemma \[piT is faithful on core\] implies that $\pi_T$ is faithful on ${\overline{{\operatorname{span}}}}\{w_pw_q^*:\mu(p)=\mu(q)\}={\operatorname{range}}\Psi^\mu$. Thus $\Psi^\mu(R^*R)=0$, and then $R=0$ since $\Psi^{\mu}$ is faithful. Hence $E$ is faithful and $(G,P)$ is amenable.

Quasi-lattice ordered HNN extensions
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Let $G$ be a group, let $A$ and $B$ be subgroups of $G$, and let $\phi:A\to B$ be an isomorphism. The group with presentation $${G^*=\langle G,t\mid t^{-1}at=\phi(a),a\in A\rangle}$$ is the *HNN extension of G* with respect to $A,B$ and $\phi$. For every HNN extension $G^*$ the *height map* is the homomorphism $\theta:G^*\to {\mathbb{Z}}$ such that $\theta(g)=0$ for all $g\in G$ and $\theta(t)=1$.

Let $c,d\in {\mathbb{Z}}$. The Baumslag-Solitar group $${\operatorname{BS}}(c,d)=\langle x,t\mid t^{-1}x^dt=x^c\rangle =\langle x,t\mid tx^c=x^dt\rangle$$ is an HNN extension of ${\mathbb{Z}}$ with respect to $A=d{\mathbb{Z}}$, $B=c{\mathbb{Z}}$ and $\phi:A\to B$ given by $\phi(dn)=cn$ for all $n\in {\mathbb{Z}}$. Then ${\mathbb{Z}}^*$ satisfies the relation $t^{-1}dt=\phi(d)=c$. Let ${\operatorname{BS}}(c,d)^+$ be the subsemigroup of ${\operatorname{BS}}(c,d)$ generated by $x$ and $t$. Spielberg showed 