C⁎-exactness and property A for group actions

Abstract

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