Abstract
We give an analogue of the Bessel inequality and we state a simple formulation of the Gruss type inequality in inner product -modules, which is a refinement of it. We obtain some further generalization of the Gruss type inequalities in inner product modules over proper -algebras and unital Banach -algebras for -seminorms and positive linear functionals.
Highlights
A proper H∗-algebra is a complex Banach ∗-algebra A, · where the underlying Banach space is a Hilbert space with respect to the inner product ·, · satisfying the properties ab, c b, a∗c and ba, c b, ca∗ for all a, b, c ∈ A
There are a positive linear functional tr on τ A and a norm τ on τ A, related to the norm of A by the equality tr a∗a τ a∗a a 2 for every a ∈ A
If Γ is a norm on an inner product A-module X, X, Γ is said to be a pre-Hilbert A-module
Summary
A proper H∗-algebra is a complex Banach ∗-algebra A, · where the underlying Banach space is a Hilbert space with respect to the inner product ·, · satisfying the properties ab, c b, a∗c and ba, c b, ca∗ for all a, b, c ∈ A. Let γ be a seminorm or a positive linear functional on A and Γ x γ x, x 1/2 x ∈. If Γ is a norm on an inner product A-module X, X, Γ is said to be a pre-Hilbert A-module. A Let A be a ∗-algebra and γ a positive linear functional or a C∗-seminorm on A. B Let A be a Hermitian Banach ∗-algebra and ρ be the Ptak function on A. D Let A be a H∗-algebra and X a semi-inner product an inner product A-module. We give an analogue of the Bessel inequality 2.7 and we obtain some further generalization and a simple form for the Gruss type inequalities in inner product modules over C∗-algebras, proper H∗-algebras, and unital Banach ∗-algebras
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