Characterizations of type 1 subdiagonal algebras and an application

Abstract

Let $\mathfrak A$ be a maximal subdiagonal algebra in a $\sigma$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $\Phi$. We consider the characterizations for $\mathfrak A$ to be a type 1 subdiagonal algebra in the sense that every right invariant subspace in noncommutative $H^2$ space is of Beurling type. As an application, we give a necessary and sufficient condition that a nest subalgebra $\rm{Alg} \mathcal N$ with an injective nest $\mathcal N$ is a type 1 subdiagonal algebra in a factor von Neumann algebra $\mathcal M$.