Ergodic actions of Abelian groups and properties of their joint actions (Q909955)

Fix a separable locally compact abelian group G and Lebesgue spaces (X,\(\mu)\) and (Y,\(\nu)\) such that \(\mu (X)=\nu (Y)=1\). Let \(\{U_ s\}_{s\in G}\) and \(\{V_ s\}_{s\in G}\) be ergodic actions of G on X and Y, respectively \((U_ s\) and \(V_ s\) are non-singular automorphisms of X and Y, respectively) and denote by \(\{U'_ s\}_{s\in G}\) and \(\{V'_ s\}_{s\in G}\) their continuous unitary representations in \(L^ 2(X,\mu)\) and \(L^ 2(Y,\nu)\), respectively. Further, let \(\{(U,V)_ s\}_{s\in G}\) denote the so called simultaneous action of \(\{U_ s\}_{s\in G}\) and \(\{V_ s\}_{s\in G}\). The main result of the paper states that \(\{(U,V)_ s\}_{s\in G}\) is ergodic, \((U,V)_ s\) preserves the measure \(\mu\) \(\times \nu\) for any \(s\in G\) and \(\{(U,V)'_ s\}_{s\in G}\) forms a basis of \(L^ 2(X\times Y,\mu \times \nu)\); moreover, the spectrum of \(\{(U,V)_ s\}_{s\in G}\) is the intersection of the spectra of \(\{U_ s\}_{s\in G}\) and \(\{V_ s\}_{s\in G}\). This is a generalization of a previous result proved by T. Hamachi and M. Osikawa.