# 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 $\mathcal{H}$.

**DOI: **http://dx.doi.org/10.7900/jot.2018feb19.2228

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

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