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 $onatracialvonNeumannalgebra$\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) $forsomepositive$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) $,westudythenoncommutativeHardyspaces$% H^{\alpha }\left( $\mathcal{M}$,\tau \right) $, then prove that the Chen-Hadwin-Shen theorem holds for $L^{\alpha }(\mathcal{M},\tau )$. The key ingredients in the proof of our result include a factorization theorem and a density theorem for $L^{\alpha }(\mathcal{M},\rho )$.