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

Let R be a real von Neumann algebra, and let U(R) be the least von Neumann algebra generated by R. We consider crossed products of U(R) and strongly continuous actions of commutative locally compact groups of *-automorphisms of U(R). We study the real structure in the von Neumann algebras dual to 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