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

Assume that $A$ and $B$ areunital $C^{*}$-algebras and $\varphi:A\rightarrow B$ is a unitalpositive linear map. We show that if $B$ is commutative, then forall $x,y \in A$ 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^{*})(\lambdaz-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.