On the Gruss Inequality for unital 2-positive linear maps

In a recent work, Moslehian and Rajic have shown that the Gruss inequality holds for unital n-positive linear maps $ϕ:\mathcal A \rightarrow B(H)$, where $\mathcal A$ is a unital C*-algebra and H is a Hilbert space, if $n \ge 3$. They also demonstrate that the inequality fails to hold, in general, if $n = 1$ and question whether the inequality holds if $n=2$. In this article, we provide an affirmative answer to this question.

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