# Journal of Operator Theory

Volume 73, Issue 1, Winter 2015 pp. 265-278.

Gruess inequality for some types of positive linear maps

**Authors**:
Jagjit Singh Matharu (1) and Mohammad Sal Moslehian (2)

**Author institution:**(1) Department of Mathematics, Bebe Nanaki University College,
Mithra, Kapurthla, Punjab, India

(2) Department of Pure Mathematics, Center of Excellence in
Analysis on Algebraic Structures (CEAAS), Ferdowsi University of
Mashhad, P.O. Box 1159, Mashhad 91775, Iran

**Summary: **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)\|\!| \leqslant \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.

**DOI: **http://dx.doi.org/10.7900/jot.2013nov20.2040

**Keywords: **operator inequality, Gruess inequality, completely
positive map, $C^*$-algebra, matrix unitarily invariant norm,
singular value

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