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