On some subalgebras of von Neumann algebras with analyticity

Let (M,α,G) be a covariant system on a locally compact Abelian group G with the totally ordered dual group which admits the positive semigroup . Let H^∞(α) be the associated analytic subalgebra of M; i.e. . Let be the analytic crossed product determined by a covariant system . We give the necessary and sufficient condition that an analytic subalgebra H^∞(α) is isomorphic to an analytic crossed product related to Landstad's theorem. We also investigate the structure of σ-weakly closed subalgebra of a continuous crossed product N?_θR which contains N?_θR₊. We show that there exists a proper σ-weakly closed subalgebra of N?_θR which contains N?_θR₊ and is not an analytic crossed product. Moreover we give an example that an analytic subalgebra is not a continuous analytic crossed product using the continuous decomposition of a factor of type III_λ(0?λ<1).