# A Beurling-Blecher-Labuschagne theorem for Haagerup noncommutative $L^p$ spaces

## TL;DR

This paper extends classical Beurling-Blecher-Labuschagne theorems to Haagerup noncommutative L^p spaces, characterizing invariant subspaces and outer operators within this noncommutative framework.

## Contribution

It establishes a Beurling-Blecher-Labuschagne type theorem for invariant subspaces of Haagerup noncommutative L^p spaces and characterizes outer operators in these spaces.

## Key findings

- Proves a Beurling-Blecher-Labuschagne theorem for Haagerup noncommutative L^p spaces.
- Provides a characterization of outer operators in Haagerup noncommutative H^p-spaces.
- Extends classical invariant subspace theory to noncommutative L^p settings.

## Abstract

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be maximal subdiagonal subalgebra of $\mathcal{M}$ and $1\le p<\8$. We prove a Beurling-Blecher-Labuschagne type theorem for $\mathcal{A}$-invariant subspaces of Haagerup noncommutative $L^p(\mathcal{M})$ and give a characterization of outer operators in Haagerup noncommutative $H^{p}$-spaces associated with $\mathcal{A}$.