Continuity of the multiplication map α on the minimal tensor product

Determine whether the multiplication map α: U⊗_min V→B(H), defined by α(a⊗b)=a·b, is continuous with respect to the minimal tensor product norm in the setting where U=\overline{π(A(X,A)⊗1)}^{WOT}, V=\overline{π(1⊗A(Y,B))}^{WOT}, and (H,π,|ψ⟩) arises from the GNS construction of a state φ on A(X,A)⊗_min A(Y,B) satisfying φ(e_a^x⊗f_b^y)=p(a,b|x,y).

Background

To prove amenability of the trace τ needed for the characterization, the authors aim to establish the continuity of the functional a⊗b↦⟨ψ|a·b|ψ⟩ by composing maps id⊗s and α on U⊗_min U^op, with s constructed in their special-case argument. They remark that α may fail to be continuous in general due to known counterexamples involving left/right regular representations of non-amenable groups.

Despite this, the specific von Neumann algebras U and V in their construction possess additional structure, leaving the general continuity of α an unresolved question crucial to completing the proof of amenability.