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 G^ which admits the positive semigroup G^+. Let H∞(α) be the associated analytic subalgebra of M; i.e. H∞(α)=x∈M∣Spα(x)⊆G^+. Let N⋊θG^+ be the analytic crossed product determined by a covariant system (N,θ,G^). We give the necessary and sufficient condition that an analytic subalgebra H∞(α) is isomorphic to an analytic crossed product N⋊θG^+ 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 IIIλ(0⩽λ<1).