A Gruss inequality for n-positive linear maps

TL;DR

This paper establishes a Gruss-type inequality for unital n-positive linear maps between C*-algebras, providing bounds on the deviation of the map from multiplicativity based on operator norm distances.

Contribution

It introduces a new inequality for n-positive maps, extending Gruss inequalities to the setting of operator algebras with explicit bounds.

Findings

Bound on (AB)-(A)(B) in terms of operator norm distances

Applicable to unital n-positive maps with n  3

Generalizes classical Gruss inequality to operator algebra context

Abstract

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