A Beurling-Blecher-Labuschagne Theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras

Abstract

In 2008, Blecher and Labuschagne extended Beurling's classical theorem to $H^\infty$-invariant subspaces of $L^p(\mathcal{M},\tau)$ for a finite von Neumann algebra $\mathcal{M}$ with a finite, faithful, normal tracial state $\tau$ when $1\le p\le \infty$. In this paper, 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 $\tau$, we prove a Beurling-Blecher-Labuschagne theorem for $H^\infty$-invariant spaces of $L^p(\mathcal{M},\tau)$ when $0<p\leq\infty$. The proof of the main result relies on proofs of density theorems for $L^p(\mathcal{M},\tau)$ and semifinite versions of several other known theorems from the finite case. 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 $0<p\leq\infty$.