A non-commutative Beurling's theorem with respect to unitarily invariant norms

Abstract

In 1967, Arveson invented a non-commutative 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's theorem on $H^{\infty }$-right invariant subspaces in a non-commutative $L^p(\mathcal{M},\tau )$ space for $1\le p\le \infty$. In the present paper, we define and study a class of norms ${\mathcal{N}}_c(\mathcal{M},\tau )$ on $\mathcal{M},$ called normalized, unitarily invariant, $\Vert \cdot {\Vert }_1$-dominating, continuous norms, which properly contains the class $\{\Vert \cdot {\Vert }_p:1\le p<\infty \}.$ For $\alpha \in {\mathcal{N}}_c(\mathcal{M},\tau ),$ we define a non-commutative $L^{\alpha }(\mathcal{M},\tau )$ space and a non-commutative $H^{\alpha }$ space. Then we obtain a version of the Blecher-Labuschagne-Beurling invariant subspace theorem on $H^{\infty }$-right invariant subspaces in a non-commutative $L^{\alpha }(\mathcal{M},\tau )$ space. Key ingredients in the proof of our main result include a characterization theorem of $H^{\alpha }$ and a density theorem for $L^{\alpha }(\mathcal{M},\tau )$.

25 pages

References in corpus (1)

• Riesz and Szegö type factorizations for noncommutative Hardy spaces