Journal of Operator Theory, ( ISI ), Volume (73), No (1), Year (2015-1) , Pages (265-278)

Title : ( Gruss inequality for some types of positive linear maps )

Authors: Jagjit Singh Matharu , Mohammad Sal Moslehian ,

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Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}_n$ for all finite values of $n$ via $|||A|||=|||A\oplus 0|||$. We show that if $\mathscr{A}$ is a $C^*$-algebra of finite dimension $k$ and $\Phi: \mathscr{A} \to \mathcal{M}_n$ is a unital completely positive map, then \begin{equation*} |||\Phi(AB)-\Phi(A)\Phi(B)||| \leq \frac{1}{4} |||I_{n}|||\,|||I_{kn}||| d_A d_B \end{equation*} for any $A,B \in \mathscr{A}$, where $d_X$ denotes the diameter of the unitary orbit $\{UXU^*: U \mbox{ is unitary}\}$ of $X$ and $I_{m}$ stands for the identity of $\mathcal{M}_{m}$. Further we get an analogous inequality for certain $n$-positive maps in the setting of full matrix algebras by using some matrix tricks. We also give a Gr\"uss operator inequality in the setting of $C^*$-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators.


, Operator inequality, Gr\"uss inequality, completely positive map; $C^*$-algebra; matrix, unitarily invariant norm, singular value.
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author = {Jagjit Singh Matharu and Sal Moslehian, Mohammad},
title = {Gruss inequality for some types of positive linear maps},
journal = {Journal of Operator Theory},
year = {2015},
volume = {73},
number = {1},
month = {January},
issn = {0379-4024},
pages = {265--278},
numpages = {13},
keywords = {Operator inequality; Gr\"uss inequality; completely positive map; $C^*$-algebra; matrix; unitarily invariant norm; singular value.},


%0 Journal Article
%T Gruss inequality for some types of positive linear maps
%A Jagjit Singh Matharu
%A Sal Moslehian, Mohammad
%J Journal of Operator Theory
%@ 0379-4024
%D 2015