covariant representation $w$; it is universal for covariant representations of $P$ in the following sense: for any covariant representation $W:P\to A$ there exists a unital homomorphism $\pi_W:C^*(G,P)\to A$ such that $\pi_W(w_p)=W_p$. It follows from that $$C^*(G,P)= \overline{{\operatorname{span}}}\{w_pw^*_q:p, q\in P\}.$$

Nica defined $(G,P)$ to be *amenable* if the homomorphism $\pi_T:C^*(G,P)\to C^*_r(G,P)$ is faithful [@Nica1992 §4.2]. He identified an equivalent condition: there exists a conditional expectation $E:C^*(G,P)\to \overline{{\operatorname{span}}}\{w_pw^*_p:p\in P\}$, and $(G,P)$ is amenable if and only if $E$ is faithful (that is, $E(a^*a)=0$ implies $a^*a=0$ for all $a\in C^*(G,P)$). Laca and Raeburn took this second condition as their definition of amenability [@LacaRaeburn1996 Definition 3.4].

Order-preserving maps and amenability
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A key technique, introduced by Laca and Raeburn in [@LacaRaeburn1996 Proposition 6.6][^2], is the use of an order-preserving homomorphism between two quasi-lattice ordered groups which preserves the least upper bound structure. Crisp and Laca called such a homomorphism a *controlled map* [@CrispLaca2007]. If $(G,P)$ and $(K,Q)$ are quasi-lattice ordered groups, $\mu:G\to K$ is a controlled map and $K$ is an amenable group, then $(G,P)$ is amenable as a quasi-lattice ordered group by [@LacaRaeburn1996 Proposition 6.6]. Motivated by work in [@HuefClarkRaeburn2015 Appendix A] we now give a weaker definition for a controlled map. We 