A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

Abstract

We prove a Beurling-Blecher-Labuschagne theorem for ${H^\infty}$-invariant spaces of ${L^p(\mathcal{M},\tau)}$ when ${0 < p \leq\infty}$, using Arveson’s non-commutative Hardy space ${H^\infty}$ in relation to a von Neumann algebra ${\mathcal{M}}$ with a semifinite, faithful, normal tracial weight ${\tau}$. Using the main result, we are able to completely characterize all ${H^\infty}$-invariant subspaces of ${L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}$, where ${\mathcal{M} \rtimes_\alpha \mathbb{Z} }$ is a crossed product of a semifinite von Neumann algebra ${\mathcal{M}}$ by the integer group ${\mathbb{Z}}$, and ${H^\infty}$ is a non-selfadjoint crossed product of ${\mathcal{M}}$ by ${\mathbb{Z}^+}$. As an example, we characterize all ${H^\infty}$-invariant subspaces of the Schatten p-class ${S^p(\mathcal{H})}$, where ${H^\infty}$ is the lower triangular subalgebra of ${B(\mathcal{H})}$, for each ${0 < p \leq\infty}$.