On some subalgebras of von Neumann algebras with analyticity

Let (M, ,G) be a covariant system on a locally compact Abelian group G with the totally ordered dual group Gˆ which admits the positive semigroup Gˆ +. Let H∞( ) be the associated analytic subalgebra of M; i.e. H∞( )= x ∈ M | Sp (x) ⊆ Gˆ + . Let N Gˆ + be the analytic crossed product determined by a covariant system (N, , Gˆ ). We give the necessary and sufficient condition that an analytic subalgebra H∞( ) is isomorphic to an analytic crossed product N Gˆ + related to Landstad’s theorem. We also investigate the structure of -weakly closed subalgebra of a continuous crossed product N R which contains N R+. We show that there exists a proper -weakly closed subalgebra of N R which contains N R+ 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 (0 <1). © 2005 Elsevier Inc. All rights reserved.