Interpolation problems in subdiagonal algebras

TL;DR

This paper investigates interpolation problems in subdiagonal algebras within von Neumann algebras, characterizes trivial ideals, and establishes a connection between type 1 subdiagonal algebras and analytic operator algebras with applications to a noncommutative Corona theorem.

Contribution

It provides a characterization of trivial ideals, links type 1 subdiagonal algebras to periodic flows, and offers a noncommutative Corona theorem variant.

Findings

Type 1 subdiagonal algebras coincide with analytic operator algebras associated with periodic flows.

A form decomposition of type 1 subdiagonal algebras is established.

A noncommutative operator-theoretic Corona theorem for type 1 subdiagonal algebras is derived.

Abstract

Let $\mathfrak A$ be a subdiagonal algebra with diagonal $\mathfrak D$ in a $\sigma$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $\Phi$. 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.