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

Abstract

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling's theorems for a continuous unitarily invariant norm on a tracial von Neumann algebra $\left(\mathcal{M},\tau \right)$ where $\alpha$ is ${\Vert \cdot \Vert }_1$-dominating with respect to $\tau$. In the paper, we first define a class of norms 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 , 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 $L^{\alpha }\left(\mathcal{M},\rho \right)$.

arXiv admin note: text overlap with arXiv:1505.03952

References in corpus (1)

• An Extension of the Chen-Beurling-Helson-Lowdenslager Theorem