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 $L^{\mathrm{\infty }}(\mathbb{T},\mu )$ and tracial finite von Neumann algebras $(\mathcal{M},\tau )$, respectively. In the paper, we study unitarily $\mathrm{\parallel }{\mathrm{\parallel }}_1$-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) $.