Subdiagonal algebras with the Beurling type invariant subspaces

TL;DR

This paper characterizes a class of maximal subdiagonal algebras in von Neumann algebras, called type 1, by their invariant subspaces, and explores their generators, factorization properties, and reflexivity of associated Toeplitz algebras.

Contribution

It identifies and characterizes type 1 subdiagonal algebras with Beurling type invariant subspaces, providing generators and analyzing their operator algebra properties.

Findings

Type 1 subdiagonal algebras are characterized by their invariant subspaces.

A Riesz type factorization theorem for non-commutative H^1 is established.

The right analytic Toeplitz algebra for these algebras with multiplicity 1 is hereditary reflexive.

Abstract

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.