Beurling type representation for certain invariant subspaces of maximal subdiagonal algebras

Abstract

Let A be a maximal subdiagonal algebra in a σ-finite von Neumann algebra M with respect to a faithful normal conditional expectation Φ. We consider certain invariant subspaces of A in the Haagerup noncommutative Lp space Lp(M) (1≤p≤∞). Suppose that φ is a faithful normal state on M with φ∘Φ=φ and {σtφ:t∈R} is the modular automorphism group of M associated with φ, we give a Beurling type representation for those right invariant subspaces of A in Lp(M) which are also invariant under {σtφ:t∈R}. Furthermore, we give an algebraic characterization for a subdiagonal algebra to be type 1 and also give a type decomposition for a central projection in M associated with the center of D.