Invariant subspaces for $H^2$ spaces of $\sigma$-finite algebras

TL;DR

This paper extends Beurling's invariant subspace theory to noncommutative $H^2$ spaces within $\sigma$-finite von Neumann algebras, establishing equivalences with maximal subdiagonality and introducing analytically conditioned algebras.

Contribution

It generalizes invariant subspace theory from finite to $\sigma$-finite von Neumann algebras and introduces the concept of analytically conditioned algebras.

Findings

Invariant subspace theory extends to $\sigma$-finite algebras.

Maximal subdiagonality characterized by equivalent properties.

Introduction of analytically conditioned algebras.

Abstract

We show that a Beurling type theory of invariant subspaces of noncommutative $H^2$ spaces holds true in the setting of subdiagonal subalgebras of $\sigma$-finite von Neumann algebras. This extends earlier work of Blecher and Labuschagne for finite algebras, and complements more recent contributions in this regard by Bekjan, and Chen, Hadwin and Shen in the finite setting, and Sager in the semifinite setting. We then also introduce the notion of an analytically conditioned algebra, and go on to show that in the class of analytically conditioned algebras this Beurling type theory is part of a list of properties which all turn out to be equivalent to the maximal subdiagonality of the given algebra.