We prove a Beurling-type 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, where M is a von Neumann algebra with a faithful, normal, semifinite tracial weight 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.
