Dual Systems of Imprimitivity

Abstract

We now discuss systems of imprimitivity based on compact group rotations. In this case there are naturally arising dual systems of imprimitivity and the two together yield considerable spectral information. First we recall the definition of a compact group rotation: Let Q ⊆ S1 be a countable infinite group. Let \(G = {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Q} _d}\) be the compact dual of Q d , where Q d is the group Q with the discrete topology. Let x0 ∈ G be the element defined by x0(q) = q for all q ∈ Q. Let τ : G → G be defined by τx = x + x0, x ∈ G. Then the system (G, τ) is called a compact group rotation.