A Generalized Beurling Theorem in Finite von Neumann Algebras

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 $α$ on $L^{\infty}(\mathbb{T},μ)$ and tracial finite von Neumann algebras $\left( \mathcal{M},τ\right) $, respectively. In the paper, we study unitarily $\|\|_{1}$-dominating invariant norms $α$ on finite von Neumann algebras. First we get a Burling theorem in commutative von Neumann algebras by defining $H^α(\mathbb{T},μ)=\overline {H^{\infty}(\mathbb{T},μ)}^{σ(L^α\left( \mathbb{T} \right),\mathcal{L}^{α^{'}}\left( \mathbb{T} \right))}\cap L^α(\mathbb{T},μ)$, 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^{α}\left(\mathcal{M},τ\right) $.