On the Gruss Inequality for unital 2-positive linear maps

TL;DR

This paper proves that the Gruss inequality holds for unital 2-positive linear maps, filling a gap in understanding the inequality's validity for this specific positivity level.

Contribution

It establishes that the Gruss inequality is valid for unital 2-positive linear maps, confirming a previously open question.

Findings

The Gruss inequality holds for unital 2-positive maps.

The result completes the understanding of the inequality for n-positive maps.

It clarifies the boundary cases between 1-positivity and higher positivity.

Abstract

In a recent work, Moslehian and Rajic have shown that the Gruss inequality holds for unital n-positive linear maps $\phi:\mathcal A \rightarrow B(H)$, 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.