A Gruss inequality for n-positive linear maps

Let $\mathscr{A}$ be a unital $C^*$-algebra and let $Φ: \mathscr{A} \to {\mathbb B}({\mathscr H})$ be a unital $n$-positive linear map between $C^*$-algebras for some $n \geq 3$. We show that $\|Φ(AB)-Φ(A)Φ(B)\| \leq Δ(A,||\cdot||)\,Δ(B,||\cdot||)$ for all operators $A, B \in \mathscr{A}$, where $Δ(C,\|\cdot\|)$ denotes the operator norm distance of $C$ from the scalar operators.

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