Bounded solutions of \(n\)-cocycle and related equations on amenable semigroups (Q1290388)

Let \(S\) be a semigroup, and let \(X\) be an Abelian group. The \((n+1)\)-coboundary generated by a map \(f:S^n \to X\) is the map \(\delta f:S^{n+1} \to X\) defined by \[ \begin{multlined}\delta f(x_1,\dots,x_{n+1}):=f(x_2,\dots,x_{n+1})+\\ +\sum_{i=1}^n (-1)^i f(x_1,\dots,x_ix_{i+1},\dots,x_{n+1})+ (-1)^{n+1}f(x_1,\dots,x_n)\end{multlined} \] for all \(x_1,\cdots,x_{n+1} \in S\). An \(n\)-cocycle is a map \(f:S^n \to X\) for which \(\delta f=0\). The main result of the paper is the following Theorem. Let \(S\) be a left-amenable semigroup, and let \(n \geq 2\) be a positive number. Then a map \(F:S^n \to \mathbb C\) is a bounded \(n\)-cocycle, if and only if \(F\) is an \(n\)-coboundary (\(F=\delta f\)) with bounded generating function \(f:S^{n+1} \to \mathbb C\). Moreover, the function \(f\) is unique up to an \((n-1)\)-coboundary with bounded generating function. The theorem above is then generalized, for \(n=3\), to the case of five unknown functions, i.e., to the following functional equation \[ F(x_2,x_3,x_4)- G(x_1x_2,x_3,x_4)+ H(x_1,x_2x_3,x_4)- K(x_1,x_2,x_3x_4)+ L(x_1,x_2,x_3)=0. \]