A Grüss inequality for n-positive linear maps

Let AA be a unital C∗C∗-algebra and let Φ:A→B(H)Φ:A→B(H) be a unital n  -positive linear map between C∗C∗-algebras for some n⩾3n⩾3. We show that‖Φ(AB)-Φ(A)Φ(B)‖⩽Δ(A,||·||)Δ(B,||·||)for all operators A,B∈AA,B∈A, where Δ(C,‖·‖)Δ(C,‖·‖) denotes the operator norm distance of C from the scalar operators.