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A Hölder-type inequality for positive functionals on \(\Phi\)-algebras. (English) Zbl 1022.06008

Summary: The main purpose of this paper is to establish with a constructive proof the following Hölder-type inequality: let \(A\) be a uniformly complete \(\Phi\)-algebra, \(T\) be a positive linear functional, and \(p,q\) be rational numbers such that \(p^{-1}+ q^{-1}=1\). Then the inequality \(T(|fg|) \leq(T(|f|^p))^{1/p} (T(|g|^q))^{1/q}\) holds for all \(f,g\in A\).

MSC:

06F25 Ordered rings, algebras, modules
47B65 Positive linear operators and order-bounded operators
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