Characterizations of type 1 subdiagonal algebras and an application

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 the characterizations for A to be a type 1 subdiagonal algebra in the sense that every right invariant subspace in noncommutative H² space is of Beurling type. As an application, we give a necessary and sufficient condition that a nest subalgebra Alg N with an injective nest N is a type 1 subdiagonal algebra in a factor von Neumann algebra M.