Improvement of the Gr\"{u}ss type inequalities for positive linear maps on $C^{*}$-algebras

Document Type : Research Articles

Authors

1 Tabriz Islamic Art University

2 Department of Mathematics, Sahand University of Technology, Tabriz, Iran

3 Department of Mathematics Sahand University of Technology

Abstract

Assume that $A$ and $B$ are
unital $C^{*}$-algebras and $\varphi:A\rightarrow B$ is a unital
positive linear map. We show that if $B$ is commutative, then for
all $x,y \in A$ and $\alpha, \beta \in \mathbb{C}$
\begin{align*}
|\varphi(xy)-\varphi(x)\varphi(y)| \leq & \left[
\varphi(|x^{*}-\alpha 1_{A}|^{2})\right]
^{\frac{1}{2}}\left[\varphi(|y-\beta1_{A}|^{2})\right]
^{\frac{1}{2}} \\ & - |\varphi(x^{*}-\alpha 1_{A})|
|\varphi(y-\beta1_{A})|.
\end{align*}
Furthermore, we prove that if $z\in A$
with $|z| =1$ and $\lambda, \mu \in \mathbb{C}$ are such that
$Re(\varphi((x^{*}-\bar{\beta}z^{*})(\alpha z-x)))\geq 0$ and
$Re(\varphi((y^{*}-\bar{\mu}z^{*})(\lambda z-y)))\geq 0$, then
\begin{center}
$|\varphi(x^{*}y)-\varphi(x^{*}z)\varphi(z^{*}y)| \leq \frac{1}{4}| \beta-\alpha | | \mu-\alpha | -$ \\
$ \left[ Re(\varphi((x^{*}-\bar{\beta}z^{*})(\alpha z-x)))\right]
^{\frac{1}{2}}\left[ Re(\varphi((y^{*}-\bar{\mu}z^{*})(\lambda
z-y)))\right] ^{\frac{1}{2}}.$
\end{center}
The presented bounds for the Gr\"{u}ss type inequalities on $C^{*}$-algebras improve the other ones in the literature under mild conditions. As an application, using our results, we give some inequalities in $L^{\infty}(\left[a,b\right])$, which refine the other ones in the literature.

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