Equivalence between quantum approximate perfection and amenable tracial state characterization for imitation games

Determine whether, for every imitation game G=(X,Y,A,B,λ) and perfect non-signalling correlation p∈C_ns(G), the following equivalence holds: (i) p∈C_qa(G); (ii) there exist a von Neumann algebra U with an amenable tracial state τ on U and a *-homomorphism ρ: C⟨X,A⟩⊗_alg C⟨Y,B⟩→U such that τ∘ρ(e_a^x⊗f_b^y)=p(a,b|x,y) for all x∈X, y∈Y, a∈A, b∈B. If the equivalence fails in general, identify the precise additional conditions under which it holds.

Background

The paper proves a sufficient condition (Theorem 3.2): if there exist a von Neumann algebra U, a *-homomorphism ρ from the algebra generated by player projections, and an amenable tracial state τ inducing the correlation, then p lies in C_qa(G). The authors propose a main theorem (Proposition P_conj) asserting an equivalence, but they have not proved it in full generality.

They discuss obstacles to establishing amenability of the candidate trace via continuity of certain tensor product maps and note that while a non-amenable group counterexample shows such continuity can fail in general, their algebras U and V have additional structure. As a result, they are uncertain whether the equivalence holds for all imitation games or if extra hypotheses are needed.