A Beurling theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras with unitarily invariant norms

Abstract

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.