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On separation theorems for subadditive and superadditive functionals. (English) Zbl 0739.39014
Let \((S,\cdot)\) be a semigroup, \(f: S\to\mathbb{R}\) a subadditive functional, \(g: S\to\mathbb{R}\) a superadditive functional and assume \(g(x)\leq f(x)\) for all \(x\in S\). The authors deal with the problem of separating \(g\) and \(f\) by means of an additive functional. They obtain results when \(S\) is weakly commutative (i.e. \(\forall x,y\in S\) \(\exists n\in\mathbb{N}\): \((xy)^{2n}=x^{2n}y^{2n})\) and/or is amenable. Some of the results are related to Hyers-Ulam stability of the Cauchy equation.
Reviewer: G.L.Forti (Milano)

MSC:
39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
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