Schwarz and Grüss type inequalities for [formula omitted]-seminorms and positive linear functionals on Banach ∗-modules

Abstract

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 ( < x , x > ) ) 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 .