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Journal of Operator Theory

Volume 75, Issue 2, Spring 2016  pp. 497-523.

A noncommutative Beurling theorem with respect to unitarily invariant norms

Authors:  Yanni Chen (1) Don Hadwin (2) Junhao Shen (3)
Author institution:(1) School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, China
(2) Department of Mathematics, University of New Hampshire, Durham, NH 03824, U.S.A.
(3) Department of Mathematics, University of New Hampshire, Durham, NH 03824, U.S.A.

Summary: In 1967, Arveson invented a noncommutative generalization of classical $H^{\infty},$ known as finite maximal subdiagonal subalgebras, for a finite von Neumann algebra $\mathcal{M}$ with a faithful normal tracial state $\tau$. In 2008, Blecher and Labuschagne proved a version of Beurling theorem on $H^{\infty}$-right invariant subspaces in a noncommutative $L^{p}% (\mathcal{M},\tau)$ space for $1\leqslant p\leqslant \infty$. In the present paper, we define and study a class of norms ${{N}}_{c}(\mathcal{M}, \tau)$ on $\mathcal{M},$ called normalized, unitarily invariant, $\Vert \cdot \Vert_{1}% $-dominating, continuous norms, which properly contains the class $\{ \Vert \cdot \Vert_{p}:1\leqslant p< \infty \}$ and the class of rearrangement invariant quasi Banach function norms studied by Bekjan. For $\alpha \in {N}% _{c}(\mathcal{M}, \tau),$ we define a noncommutative $L^{\alpha}% ({\mathcal{M}},\tau)$ space and a noncommutative $H^{\alpha}$ space. Then we obtain a version of the Blecher--Labuschagne--Beurling invariant subspace theorem on $H^{\infty}$-right invariant subspaces in\break $L^{\alpha}({\mathcal{M}},\tau)$ spaces and $H^\alpha$ spaces. Key ingredients in the proof of our main result include a characterization theorem of $H^{\alpha}$ and a density theorem for $L^{\alpha}(\mathcal{M},\tau)$.

Keywords: normalized, unitarily invariant, $\Vert \cdot \Vert_{1}$-dominating, continuous norm, maximal subdiagonal algebra, dual space, Beurling theorem, noncommutative Hardy space

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