$\mathrm{C}^*$-exactness and property A for group actions

Abstract: For an action of a discrete group $\Gamma$ on a set $X$, we show that the Schreier graph on $X$ has property A if and only if the permutation representation on $\ell_2X$ generates an exact $\mathrm{C}^*$-algebra. This is well known in the case of the left regular action on $X=\Gamma$ as the equivalence of $\mathrm{C}^*$-exactness and property A of its Cayley graph. This also generalizes Sako's theorem, which states that exactness of the uniform Roe algebra $\mathrm{C}^*_{\mathrm{u}}(X)$ characterizes property A of $X$ when $X$ is uniformly locally finite.