Subdiagonal algebras with the Beurling type invariant subspaces

Let $\mathfrak{A}$ be a maximal subdiagonal algebra in a $\sigma$-finite von Neumann algebra $\mathcal{M}$. If every right invariant subspace of $\mathfrak A$inthenon-commutativeHardyspace$H^2$isofBeurlingtype,thenwesay$\mathfrak A$ to be type 1. We determine generators of these algebras and consider a Riesz type factorization theorem for the non-commutative $H^1$ space. We show that the right analytic Toeplitz algebra on the non-commutative Hardy space $H^p$ associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive.