$\theta$ be a tracial state of a unital C$^*$-algebra $B$. The following statements are equivalent:

1.  The tracial state $\theta$ is amenable,

2.  \[TheoremAmenableTraceConditionNet\] There exists a net of unital completely positive maps $\Psi_j \colon B \rightarrow \mathbb{M}(n(j))$ satisfying that $\theta(b) = \lim \mathrm{tr} \circ \Psi_j(b)$ and $\lim \| \Psi_j(ab) - \Psi_j (a) \Psi_j (b) \|_{\mathrm{tr}, 2} = 0$, for every $a, b \in B$, where $\mathrm{tr}$ stands for the normalized trace of $\mathbb{M}(n(j))$,

3.  For every faithful representation $B \subseteq \mathbb{B} (\mathcal{H})$, there exists a unital completely positive map $\Psi \colon \mathbb{B} (\mathcal{H}) \rightarrow \pi_\theta(B)^{\prime\prime}$ whose restriction to $B$ is $\pi_\theta$.

Amenability of $\theta$ passes to that on the algebra $\pi_\theta(B)$, if $B$ is locally reflexive.

\[PropositionQuotient\] Let $\theta$ be an amenable trace of a unital C$^*$-algebra $B$. If $B$ is locally reflexive, then the state $\tau = \langle \cdot \xi_\theta, \xi_\theta \rangle$ is an amenable trace of $\pi_\theta(B)$.

Since $B$ is locally reflexive, by Proposition \[PropositionLocalSplit\], the sequence $0 \rightarrow \mathrm{Ker}(\pi_\theta) \rightarrow B \rightarrow \pi_\theta(B) \rightarrow 0$ is a locally split extension. By [@OzawaBook Proposition 6.3.5 (4)], $\tau$ is an amenable trace of $\pi_\theta(B)$.

For a generalized box space $X = \bigsqcup_{m = 1}^\infty X_m$, the uniform Roe 