Improvement of the Grüss type inequalities for positive linear maps on C∗-algebras

Assume that A and B are unital C∗ -algebras and φ: A → B is a unital positive linear map. We show that if B is commutative, then for all x, y ∈ A and α, β ∈ C |φ(xy) − φ(x)φ(y)| ≤ [ φ(|x∗ − α1A|²) ] 1/2 [ φ(|y − β1A| 2 ) ] 1/2 − |φ(x∗ − α1A)||φ(y − β1A)|. Furthermore, we prove that if z ∈ A with |z| = 1 and λ, µ ∈ C are such that Re(φ((x∗ −β¯z∗)(αz −x))) ≥ 0 and Re(φ((y∗ −µ¯z∗)(λz − y))) ≥ 0, then |φ(x∗ y) − φ(x∗ z)φ(z∗ y)| ≤ 1 4 |β − α||µ − α|− [ Re(φ((x ∗ − β¯z∗)(αz − x)))] 1/2 [Re(φ((y∗ − µ¯z∗)(λz − y)))] 1/2 . The presented bounds for the Grüss 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 ∞([a, b]), which refine the other ones in the literature.