An Extension of the Beurling-Chen-Hadwin-Shen Theorem for Noncommutative Hardy Spaces Associated with Finite von Neumann Algebras

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling's theorems for a continuous unitarily invariant norm $% \alpha $ on a tracial von Neumann algebra $\left( \mathcal{M},\tau \right) $ where $\alpha $ is $\left\Vert \cdot \right\Vert _{1}$-dominating with respect to $\tau $. In the paper, we first define a class of norms $% N_{\Delta }\left( \mathcal{M},\tau \right) $ on $\mathcal{M}$, called determinant, normalized, unitarily invariant continuous norms on $\mathcal{M}$. If $\alpha \in N_{\Delta }\left( \mathcal{M},\tau \right) $, then there exists a faithful normal tracial state $\rho $ on $\mathcal{M}$ such that $\rho \left( x\right) =\tau \left( xg\right) $ for some positive $g\in L^{1}\left( \mathcal{Z},\tau \right) $ and the determinant of $g$ is positive. For every $\alpha \in N_{\Delta }\left( \mathcal{M},\tau \right) $, we study the noncommutative Hardy spaces $% H^{\alpha }\left( \mathcal{M},\tau \right) $, then prove that the Chen-Hadwin-Shen theorem holds for $L^{\alpha }\left( \mathcal{M},\tau \right) $. The key ingredients in the proof of our result include a factorization theorem and a density theorem for $L^{\alpha }\left( \mathcal{M},\rho \right) $.