Operator algebraic characterization of the noncommutative Poisson boundary

The paper’s Theorem A is clearly stated and proved: starting from Θ̂: M→B with Θ̂|M=id, it defines Θ=EB∘Θ̂, uses the unique µ-stationarity to force Θ∘ι=id on C(B), upgrades ι to a ∗-hom into mult(Θ), obtains Θ(mult(Θ))=L∞(B), and then invokes Lemma 2 to conclude Θ̂(mult(Θ̂))=B and that Θ̂ is a conditional expectation, exactly as claimed . By contrast, the candidate solution hinges on an unconstructed comparison map Φ with P̂µ∘Φ=P̂µ∘Θ̂ and on applying Choi’s multiplicative-domain argument to this composition without proving that the composed map is a ∗-homomorphism; it also misaligns several arrows (e.g., expectations and module maps) relative to the paper’s construction. These gaps make the model’s proof incorrect/incomplete.