Takesaki's duality theorem and continuous decomposition for real factors of type III

Let $R$ be a real von Neumann algebra, and let $\mathcal U(R)$ be the least von Neumann algebra generated by $R$. We consider crossed products of $\mathcal U(R)$ and strongly continuous actions of commutative locally compact groups of $^*$-automorphisms of $\mathcal U(R)$. We study the real structure in the von Neumann algebras dual to $\mathcal U(R)$ (in the sense of the Takesaki duality for crossed products). We obtain a theorem on the continuous decomposition of real factors of type III.