Linear and Multilinear Algebra, ( ISI ), Volume (65), No (7), Year (2017-7) , Pages (1386-1401)

Title : ( Gruss type inequalities for positive linear maps on C*-algebras )

Authors: Ali Dadkhah , Mohammad Sal Moslehian ,

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Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^*$-algebras and let for $C\in\mathcal{A},\ \Gamma_C=\{\gamma \in \mathbb{C} : \|C-\gamma I\|=\inf_{\alpha\in \mathbb{C}} \|C-\alpha I\|\}$. We prove that if $\Phi :\mathcal{A} \longrightarrow \mathcal{B}$ is a unital positive linear map, then \begin{eqnarray*} \big|\Phi(AB)-\Phi(A)\Phi(B)\big| \leq \big\|\Phi(|A^*-\zeta I|^2)\big\|^\frac{1}{2} \big[\Phi(|B-\xi I|^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|\tau(TAB)-\tau(TA)\tau(TB)\big|\leq \|A-\zeta I\|_p\|B-\xi I\|_q\|T\|_r \ \ (p,q\geq 4, r\geq 2) \end{eqnarray*} and \begin{eqnarray*} \big|\tau(TAB)-\tau(TA)\tau(TB)\big|\leq \|A-\zeta I\|_p\|B-\xi I\|_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.

Keywords

, Gr\"{u}ss inequality, Positive linear map, Noncommutative probability space, Trace, Hilbert $C^*$-module
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@article{paperid:1064484,
author = {Dadkhah, Ali and Sal Moslehian, Mohammad},
title = {Gruss type inequalities for positive linear maps on C*-algebras},
journal = {Linear and Multilinear Algebra},
year = {2017},
volume = {65},
number = {7},
month = {July},
issn = {0308-1087},
pages = {1386--1401},
numpages = {15},
keywords = {Gr\"{u}ss inequality; Positive linear map; Noncommutative probability space; Trace; Hilbert $C^*$-module},
}

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%0 Journal Article
%T Gruss type inequalities for positive linear maps on C*-algebras
%A Dadkhah, Ali
%A Sal Moslehian, Mohammad
%J Linear and Multilinear Algebra
%@ 0308-1087
%D 2017

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