A Generalized Beurling Theorem in Finite von Neumann Algebras

TL;DR

This paper extends Beurling's theorem to finite von Neumann algebras using unitarily invariant norms, providing a generalized framework for both commutative and noncommutative cases.

Contribution

It introduces a generalized Beurling theorem for finite von Neumann algebras under unitarily norms, expanding previous results to broader norm classes.

Findings

Established a Beurling theorem in commutative von Neumann algebras.

Proved a similar Beurling theorem in noncommutative von Neumann algebras.

Developed key factorization and density theorems for $L^{ ext{alpha}}( ext{M}, au)$.

Abstract

In 2016 and 2017, Haihui Fan, Don Hadwin and Wenjing Liu proved a commutative and noncommutative version of Beurling's theorems for a continuous unitarily invariant norm $\alpha $ on $L^{\infty}(\mathbb{T},\mu)$ and tracial finite von Neumann algebras $\left( \mathcal{M},\tau \right) $, respectively. In the paper, we study unitarily $\|\|_{1}$-dominating invariant norms $\alpha $ on finite von Neumann algebras. First we get a Burling theorem in commutative von Neumann algebras by defining $H^{\alpha}(\mathbb{T},\mu)=\overline {H^{\infty}(\mathbb{T},\mu)}^{\sigma(L^{\alpha}\left( \mathbb{T} \right),\mathcal{L}^{\alpha^{'}}\left( \mathbb{T} \right))}\cap L^{\alpha}(\mathbb{T},\mu)$, then prove that the generalized Beurling theorem holds. Moreover, we get similar result in noncommutative case. The key ingredients in the proof of our result include a factorization theorem and a density theorem for $L^{\alpha }\left(\mathcal{M},\tau \right) $.