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 ۱_{A}|^{۲})\right]^{\frac{۱}{۲}}\left[\varphi(|y-\beta۱_{A}|^{۲})\right]^{\frac{۱}{۲}} \\ & - |\varphi(x^{*}-\alpha ۱_{A})||\varphi(y-\beta۱_{A})|.\end{align*}Furthermore, we prove that if z\in Awith |z| =۱ and \lambda, \mu \in \mathbb{C} are such thatRe(\varphi((x^{*}-\bar{\beta}z^{*})(\alpha z-x)))\geq ۰ andRe(\varphi((y^{*}-\bar{\mu}z^{*})(\lambda z-y)))\geq ۰, then\begin{center}|\varphi(x^{*}y)-\varphi(x^{*}z)\varphi(z^{*}y)| \leq \frac{۱}{۴}| \beta-\alpha | | \mu-\alpha | - \\ \left[ Re(\varphi((x^{*}-\bar{\beta}z^{*})(\alpha z-x)))\right]^{\frac{۱}{۲}}\left[ Re(\varphi((y^{*}-\bar{\mu}z^{*})(\lambdaz-y)))\right] ^{\frac{۱}{۲}}.\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.

Keywords:

C^{*}-algebras , Gr\"{u}ss type inequalities , Positive linear map