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Fréchet’s equation and Hyers theorem on noncommutative semigroups. (English) Zbl 0656.39005

Let G be a semigroup with identity, H a linear space over the rationals and \(n\geq 0\) a fixed integer. It is known that if G is abelian a function f: \(G\to H\) is a solution of Fréchet’s functional equation (1) \(\Delta^{n+1}_{y_ 1...y_{n+1}}f(x)=0\) \((\Delta_ yf(x)=f(xy)- f(x),\) \(\Delta^{n+1}_{y_ 1...y_{n+1}}=\Delta_{y_{n+1}}\cdot \Delta_{y_ n}...\Delta_{y_ 1})\) if and only if it can be expressed as a sum of the diagonalizations of multiadditive symmetric functions of at most n-th degree [cf. D. Z. Djokovic, ibid. 22, 189-198 (1969; Zbl 0187.399)]. In the first part of the paper under review it is proved that this result remains true for non-commutative semigroups, under the additional assumption \(f(txy)=f(tyx)\) for all x, y in G.
In the second part it is proved the following. Let G be an amenable semigroup with identity and f: \(G\to {\mathbb{C}}^ a \)function for which \(\Delta^{n+1}_{y_ 1...y_{n+1}}f(x)\) is uniformly bounded. Then there exists a function P: \(G\to {\mathbb{C}}\) satisfying (1) for which f-P is bounded. This is a generalization of a theorem proved by D. H. Hyers when G is abelian [Pac. J. Math. 11, 591-602 (1961; Zbl 0099.105)].
Reviewer: G.L.Forti

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
43A07 Means on groups, semigroups, etc.; amenable groups
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