Schwarz and Grüss type inequalities for C∗-seminorms and positive linear functionals on Banach ∗-modules

Let A be a unital Banach ∗-algebra, γ a C∗-seminorm or a positive linear functional on A and X be a semi-inner product A-module. We define a real function Γ on X by Γ(x)=(γ())1/2 and show that the Schwarz inequality holds, therefore (X,Γ) is a semi-Hilbert A-module. We also obtain some Grüss type inequalities for C∗-seminorms and positive linear functionals on A.