Grüss type inequalities for positive linear maps on $C^{\ast }$-algebras

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^{\ast }$-algebras and let for $C\in \mathcal{A},{\Gamma }_C=\{\gamma \in \mathbb{C}:\Vert C-\gamma I\Vert ={\inf }_{\alpha \in \mathbb{C}}\Vert C-\alpha I\Vert \}$. We prove that if $\Phi :\mathcal{A}⟶\mathcal{B}$ is a unital positive linear map, then \begin{eqnarray*} \big|Φ(AB)-Φ(A)Φ(B)\big| \leq \big\|Φ(|A^*-ζI|^2)\big\|^\frac{1}{2} \big[Φ(|B-ξI|^2)\big]^\frac{1}{2} \end{eqnarray*} for all $A,B\in \mathcal{A},\zeta \in {\Gamma }_A$ and $\xi \in {\Gamma }_B.$\\ In addition, we show that if $(\mathcal{A},\tau )$ is a noncommutative probability space and $T\in \mathcal{A}$ is a density operator, then \begin{eqnarray*} \ \ \big|τ(TAB)-τ(TA)τ(TB)\big|\leq \|A-ζI\|_p\|B-ξI\|_q\|T\|_r \ \ (p,q\geq 4, r\geq 2) \end{eqnarray*} and \begin{eqnarray*} \big|τ(TAB)-τ(TA)τ(TB)\big|\leq \|A-ζI\|_p\|B-ξI\|_q\|T\| \ \ \ \ (p,q\geq 2)\ \ \ \ \ \end{eqnarray*} for every $A,B\in \mathcal{A}$ and $\zeta \in {\Gamma }_A,\xi \in {\Gamma }_B$. Our results generalize the corresponding results for matrices to operators on spaces of arbitrary dimension.

15 pages; to appear in Linear Multilinear Algebra