${\mathrm{C}}^{\ast }$-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 ${\ell }_{2}X$ generates an exact ${\mathrm{C}}^{\ast }$-algebra. This is well known in the case of the left regular action on $X=\Gamma$ 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.

11 pages, Comments are welcome

References in corpus (4)

• Amenable actions and exactness for discrete groups
• Property A and the operator norm localization property for discrete metric spaces
• A generalization of expander graphs and local reflexivity of uniform Roe algebras
• Finite-dimensional approximation properties for uniform Roe algebras