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

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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 (M,τ) where α is · ₁-dominating with respect to τ. In the paper, we first define a class of norms % NΔ(M,τ) on M, called determinant, normalized, unitarily invariant continuous norms on M. If α∈ NΔ(M,τ), then there exists a faithful normal tracial state ρ on M such that ρ(x) =τ(xg) for some positive g∈ L¹(Z,τ) and the determinant of g is positive. For every α∈ NΔ(M,τ), we study the noncommutative Hardy spaces % Hα(M,τ), then prove that the Chen-Hadwin-Shen theorem holds for Lα(M,τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for Lα(M,ρ).