A Beurling-Chen-Hadwin-Shen Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras with Unitarily Invariant Norms

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We introduce a class of unitarily invariant, locally ·₁-dominating, mutually continuous norms with repect to τ on a von Neumann algebra M with a faithful, normal, semifinite tracial weight τ. We prove a Beurling-Chen-Hadwin-Shen theorem for H∞-invariant spaces of Lα(M,τ), where α is a unitarily invariant, locally ·₁-dominating, mutually continuous norm with respect to τ, and H∞ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the H∞-invariant subspaces of a noncommutative Banach function space I(τ) with the norm ·ₑ on M, the crossed product of a semifinite von Neumann algebra by an action β, and B(H) for a separable Hilbert space H.