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

For an action of a discrete group $\mathrm{\Gamma }$ on a set $X$, we show that the Schreier graph on $X$ has property A if and only if the permutation representation on ${\mathrm{\ell }}_{2}X$ generates an exact ${\mathrm{C}}^{\ast }$-algebra. This is well known in the case of the left regular action on $X=\mathrm{\Gamma }$ as the equivalence of ${\mathrm{C}}^{\ast }$-exactness and property A of its Cayley graph. This also generalizes Sako’s theorem, which states that exactness of the uniform Roe algebra ${\mathrm{C}}_{\mathrm{u}}^{\ast }(X)$ characterizes property A of $X$ when $X$ is uniformly locally finite.