  Let $\mathfrak A$ be a maximal subdiagonal algebra in a $\sigma$-finite von
Neumann algebra $\mathcal M$. If every right invariant subspace of $\mathfrak
A$ in the non-commutative Hardy space $H^2$ is of Beurling type, then we say
$\mathfrak A$ 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.
