On a class of subdiagonal algebras

TL;DR

This paper introduces new classes of operator algebras, semi-$\sigma$-finite subdiagonal and Riesz approximable, expanding the framework for noncommutative Hardy space theory based on Arveson's subdiagonal algebras.

Contribution

It develops the theory of these new classes and explores their properties, broadening the scope of noncommutative Hardy spaces.

Findings

Defined semi-$\sigma$-finite subdiagonal algebras.

Defined Riesz approximable algebras.

Extended the noncommutative Hardy space framework.

Abstract

We investigate some new classes of operator algebras which we call semi-$\sigma$-finite subdiagonal and Riesz approximable. These constitute the most general setting to date for a noncommutative Hardy space theory based on Arveson's subdiagonal algebras. We develop this theory and study the properties of these new classes.