Improvement of the Gr\

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

φ : A \to B

is a unital
positive linear map. We show that if

B

is commutative, then for
all

x, y \in A

and

α, β \in C

\begin{aligned} | φ (x y) - φ (x) φ (y) | \leq & {[φ (| x^{*} - α 1_{A} |^{2})]}^{\frac{1}{2}} {[φ (| y - β 1_{A} |^{2})]}^{\frac{1}{2}} \\ - | φ (x^{*} - α 1_{A}) | | φ (y - β 1_{A}) | . \end{aligned}

Furthermore, we prove that if

z \in A

with

| z | = 1

and

λ, μ \in C

are such that

R e (φ ((x^{*} - \bar{β} z^{*}) (α z - x))) \geq 0

and

R e (φ ((y^{*} - \bar{μ} z^{*}) (λ z - y))) \geq 0

, then

Unknown environment '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} ([a, b])

, which refine the other ones in the literature.

Keywords