Gr\"uss type inequalities for positive linear maps on $C^*$-algebras
Ali Dadkhah, Mohammad Sal Moslehian

TL;DR
This paper extends Grüss type inequalities to positive linear maps on $C^*$-algebras, providing bounds for deviations of maps from multiplicativity, with applications to noncommutative probability spaces and operator theory.
Contribution
It introduces generalized Grüss inequalities for positive linear maps on $C^*$-algebras, broadening previous matrix-based results to infinite-dimensional operator settings.
Findings
Derived bounds for positive linear maps on $C^*$-algebras.
Extended inequalities to noncommutative probability spaces.
Generalized matrix results to operators of arbitrary dimension.
Abstract
Let and be two unital -algebras and let for . We prove that if 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 and \\ In addition, we show that if is a noncommutative probability space and 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…
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