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

Abstract

We introduce a class of unitarily invariant, locally $\Vert \cdot {\Vert }_1$-dominating, mutually continuous norms with repect to $\tau$ on a von Neumann algebra $\mathcal{M}$ with a faithful, normal, semifinite tracial weight $\tau$. We prove a Beurling-Chen-Hadwin-Shen theorem for $H^{\infty }$-invariant spaces of $L^{\alpha }(\mathcal{M},\tau )$, where $\alpha$ is a unitarily invariant, locally $\Vert \cdot {\Vert }_1$-dominating, mutually continuous norm with respect to $\tau$, and $H^{\infty }$ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the $H^{\infty }$-invariant subspaces of a noncommutative Banach function space $\mathcal{I}(\tau )$ with the norm $\Vert \cdot {\Vert }_E$ on $\mathcal{M}$, the crossed product of a semifinite von Neumann algebra by an action $\beta$, and $B(\mathcal{H})$ for a separable Hilbert space $\mathcal{H}$.

References in corpus (2)

• Riesz and Szegö type factorizations for noncommutative Hardy spaces
• On the maximality of subdiagonal algebras