Interpolation problems in subdiagonal algebras

Abstract

Let $\mathfrak{A}$ be a subdiagonal algebra with diagonal $\mathfrak{D}$ in a $\ddot{I}ƒ$-finite von Neumann algebra $\mathcal{M}$ with respect to a faithful normal conditional expectation $\hat{I}¦$. We mainly consider the interpolation problem in $\mathfrak{A}$ with the universal factorization property. We determine when a finitely generated left ideal in $\mathfrak{A}$ is trivial. By constructing a periodic flow on $\mathcal{M}$ according to a type 1 subdiagonal algebra, we show that type 1 subdiagonal algebras coincide with analytic operator algebras associated with periodic flows in von Neumann algebras. This enables us to present a form decomposition of a type 1 subdiagonal algebra. As an application, we deduce a noncommutative operator-theoretic variant of the Corona theorem for type 1 subdiagonal algebras.