A Gruss inequality for n-positive linear maps

Abstract

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

arXiv comment7 pages; to appear in Linear Algebra Appl. (LAA)

References in corpus (1)

• A treatment of the Cauchy--Schwarz inequality in $C^{\ast }$-modules