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 $\alpha$ on $L^{\infty }(\mathbb{T},\mu )$ and tracial finite von Neumann algebras $\left(\mathcal{M},\tau \right)$, respectively. In the paper, we study unitarily $\Vert {\Vert }_1$-dominating invariant norms $\alpha$ on finite von Neumann algebras. First we get a Burling theorem in commutative von Neumann algebras by defining $H^{\alpha }(\mathbb{T},\mu )={\overline{H^{\infty }(\mathbb{T},\mu )}}^{\sigma (L^{\alpha }\left(\mathbb{T}\right),{\mathcal{L}}^{{\alpha }^{{}^{\prime }}}\left(\mathbb{T}\right))}\cap L^{\alpha }(\mathbb{T},\mu )$, 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 $L^{\alpha }\left(\mathcal{M},\tau \right)$.