On the GrÃœss inequality for unital 2-positive linear maps

In a recent work, Moslehian and Rajić have shown that the Grüss inequality holds for unital n-positive linear maps φ:A →B(H), where A is a unital C∗-algebra and H is a Hilbert space, if n ≥ 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.