A Generalized Beurling Theorem in Finite von Neumann Algebras

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 $L^{\alpha }\left(\mathcal{M},\tau \right)$.