Duality for crossed products of von Neumann algebras by locally compact groups

The duality for crossed products of von Neumann algebras by locally compact abelian groups has been obtained by Takesaki [4]. We shall generalize this result to a locally compact (not necessarily abelian) group by using the Fourier algebra in place of the dual group. Let G denote a locally compact group with a right invariant Haar measure dt, and M denote a von Neumann algebra over a Hubert space H. By an action of G on M we mean a homomorphism o: t GG [-+ otE Aut(Af) such that for each x G M the mapping t G G h> ot(x) is a-strongly* continuous. Let {7Ta, X} be a covariant representation of {Af, o] on H ® L (G) defined by