Local reflexivity of the permutation C*-algebra vs Property A

Determine whether local reflexivity of the C*-algebra C*(λ^X(Γ)), generated by the permutation representation λ^X: Γ → B(ℓ^2(X)) arising from a discrete action Γ ↷ X, characterizes property A of the associated Schreier graph |Γ ⋉ X|; equivalently, ascertain whether X has property A if and only if C*(λ^X(Γ)) is locally reflexive.

Background

Property A for uniformly locally finite metric or coarse spaces admits several operator-algebraic characterizations, such as nuclearity and, under additional assumptions, exactness or local reflexivity of the uniform Roe algebra C*u(X). For discrete groups Γ, property A of the Cayley graph is equivalent to exactness of C*red(Γ).

This paper proves that for a discrete group Γ acting on a set X, the Schreier graph |Γ ⋉ X| has property A if and only if the C*-algebra C*(λ^X(Γ)) generated by the permutation representation on ℓ^2(X) is exact. By contrast with exactness, the status of local reflexivity for C*(λ^X(Γ)) as a characterization of property A is explicitly identified as unresolved in the introduction, despite analogous results of Sako for the uniform Roe algebra C*u(X).