Spectra of ergodic transformations (Q1393703)

Let \(\mathcal R\) be a von Neumann algebra acting on a Hilbert space \(\mathcal H\) and \(G\) a locally compact abelian group. Suppose \(g\to U_g\), is a strongly continuous unitary representation of \(G\) on \(\mathcal H\) implementing an ergodic group of automorphisms \(\alpha_g\) of \(\mathcal R\). By Stone's theorem there is a projection valued measure \(P_\lambda\) on the dual of \(G\) on \(\mathcal H\) such that \(U_g = \int \overline{(\lambda,g)}\,dP\). The purpose of the paper is to apply the theory of spectral subspaces for the representation \(\alpha: G\to\Aut\mathcal R\) to study the support of the measure \(P\), or if \(G\) is cyclic generated by \(g\), the spectrum of \(U_g\). In the latter case if \(\mathcal R\) is not both finite dimensional and abelian, the spectrum of \(U_g\) is the unit circle.