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.