``Pexiderized'' homogeneity almost everywhere (Q854084)

Assume that \(G\) and \(H\) are groups with zeros, \(X\) is a nontrivial \(G\)-space and \(Y\) is a \(H\)-space. Suppose that \({\mathcal I}(G)\), \({\mathcal I}(X)\) and \({\mathcal I}(G\times X)\) are conjugated proper linearly invariant ideals in \(G\), \(X\) and \(G\times X\), respectively. The author considers a homogeneity equation postulated everywhere: \[ F_1(\alpha x)=\phi(\alpha)F_2(x)\quad \text{ for}\;{\mathcal I}(G\times X)\text{ --almost all}\;(\alpha,x)\in G\times X \eqno{(1)} \] with unknown functions \(\phi\colon G\to H\), \(F_1,F_2\colon X\to Y\). It is shown that if (1) is satisfied, then (apart from trivial cases) there exist a homomorphism \(\tilde{\phi}\colon G\to H\) and a constant \(a\in H^*\) such that \[ \phi=a\tilde{\phi}\quad {\mathcal I}(G)\text{ --almost everywhere in}\;G \] and there exists a mapping \(F\colon X\to Y\) such that \[ F(\alpha x)=\tilde{\phi}(\alpha)F(x),\quad \alpha\in G^*,\;x\in X \] and \[ F_1=aF,\;F_2=F\quad {\mathcal I}(X)\text{ --almost everywhere in}\;X. \] It generalizes the author's previous result for the case \(F_1=F_2\) [Acta Math. Hungar. 113, 73--83 (2006)].