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Journal of Operator Theory

Volume 82, Issue 1, Summer 2019  pp. 49-78.

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

Authors:  Wenjing Liu (1), Lauren Sager (2)
Author institution:(1) Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, U.K.
(2) Department of Mathematics, Saint Anselm College, Manchester, NH 03102, U.K.

Summary: We prove a Beurling-type theorem for $H^\infty$-invariant spaces of $L^\alpha(\mathcal{M},\tau)$, where $\alpha$ is a unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norm with respect to $\tau$, where $\mathcal{M}$ is a von Neumann algebra with a faithful, normal, semifinite tracial weight $\tau$, 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 $\|\cdot\|_{E }$ 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 $\mathcal{H}$.

Keywords: Beurling theorem, semifinite von Neumann algebra, crossed products of von Neumann algebras, invariant subspaces, Banach function spaces

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