[Paper Review] A non-commutative Beurling's theorem with respect to unitarily invariant norms

Yanni Chen, Don Hadwin+1 more|arXiv (Cornell University)|May 15, 2015

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper extends Beurling's theorem to non-commutative $L^\alpha$ spaces over finite von Neumann algebras by introducing a new class of normalized, unitarily invariant, $\|\cdot\|_1$-dominating, continuous norms $\alpha$ on $\mathcal{M}$. It establishes a non-commutative Beurling-type theorem for $H^\infty$-right invariant subspaces in $L^\alpha(\mathcal{M},\tau)$, showing such subspaces are generated by projections or unitaries, generalizing prior results for $L^p$-spaces.

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 $τ$. In 2008, Blecher and Labuschagne proved a version of Beurling's theorem on $H^\infty$-right invariant subspaces in a non-commutative $L^{p}(\mathcal M,τ)$ space for $1\le p\le \infty$. In the present paper, we define and study a class of norms ${\mathcal{N}}_{c}(\mathcal M, τ)$ on $\mathcal{M},$ called normalized, unitarily invariant, $\Vert \cdot \Vert_{1}$-dominating, continuous norms, which properly contains the class $\{ \Vert \cdot \Vert_{p}:1\leq p< \infty \}.$ For $α\in \mathcal{N}_{c}(\mathcal M, τ),$ we define a non-commutative $L^{α}({\mathcal{M}},τ)$ space and a non-commutative $H^α$ space. Then we obtain a version of the Blecher-Labuschagne-Beurling invariant subspace theorem on $H^\infty$-right invariant subspaces in a non-commutative $L^{α}({\mathcal{M}},τ)$ space. Key ingredients in the proof of our main result include a characterization theorem of $H^α$ and a density theorem for $L^α(\mathcal M,τ)$.

Motivation & Objective

• To generalize Beurling's theorem on invariant subspaces from classical and non-commutative $L^p$-spaces to a broader class of norms.
• To define and study a new class of norms $\mathcal{N}_c(\mathcal{M},\tau)$ that properly extend $\|\cdot\|_p$ for $1 \leq p < \infty$.
• To construct non-commutative $L^\alpha(\mathcal{M},\tau)$ and $H^\alpha$ spaces for such norms.
• To establish a non-commutative Beurling theorem for $H^\infty$-right invariant subspaces in $L^\alpha(\mathcal{M},\tau)$, extending the Blecher-Labuschagne result.

Proposed method

• Introduce the class $\mathcal{N}_c(\mathcal{M},\tau)$ of normalized, unitarily invariant, $\|\cdot\|_1$-dominating, continuous norms on a finite von Neumann algebra $\mathcal{M}$ with a faithful normal tracial state $\tau$.
• Define non-commutative $L^\alpha(\mathcal{M},\tau)$ and $H^\alpha$ spaces using such norms.
• Use a characterization theorem of $H^\alpha$ to analyze the structure of $H^\infty$-right invariant subspaces.
• Establish a density theorem for $L^\alpha(\mathcal{M},\tau)$ to ensure completeness and compatibility with the norm structure.
• Apply the structure theorem for $H^\infty$-right invariant subspaces in $L^\alpha(\mathcal{M},\tau)$, decomposing them into direct sums involving projections and unitaries.
• Derive corollaries for doubly invariant and simply invariant subspaces, showing they are generated by projections or unitaries.

Experimental results

Research questions

• RQ1 Can Beurling's theorem on $H^\infty$-invariant subspaces be extended beyond $L^p$-spaces to a broader class of non-commutative $L^\alpha$-spaces defined by new norms?
• RQ2 What properties must a norm on a finite von Neumann algebra satisfy to support a non-commutative Beurling theorem for $H^\infty$-right invariant subspaces?
• RQ3 How do the structure and classification of $H^\infty$-right invariant subspaces in $L^\alpha(\mathcal{M},\tau)$ depend on the choice of norm $\alpha$?
• RQ4 Under what conditions does a closed $H^\infty$-right invariant subspace in $L^\alpha(\mathcal{M},\tau)$ decompose into a direct sum of subspaces generated by partial isometries and projections?
• RQ5 Can the generalized Beurling theorem be applied to special cases such as doubly invariant or simply invariant subspaces, and what do the generators look like?

Key findings

• The paper defines a new class of norms $\mathcal{N}_c(\mathcal{M},\tau)$ that properly contains $\|\cdot\|_p$ for $1 \leq p < \infty$, enabling the construction of non-commutative $L^\alpha(\mathcal{M},\tau)$ and $H^\alpha$ spaces.
• A characterization theorem of $H^\alpha$ is established, which is essential for analyzing $H^\infty$-right invariant subspaces in $L^\alpha(\mathcal{M},\tau)$.
• A density theorem for $L^\alpha(\mathcal{M},\tau)$ is proven, ensuring the space is well-behaved under the given norm.
• The main result, Theorem 5.7, shows that any closed $H^\infty$-right invariant subspace $\mathcal{W}$ of $L^\alpha(\mathcal{M},\tau)$ decomposes as $\mathcal{Z} \oplus^{\text{col}} \bigoplus^{\text{col}}_{i\in\mathcal{I}} u_i H^\alpha$, where $u_i$ are partial isometries in $\mathcal{M} \cap \mathcal{W}$ with orthogonal initial projections.
• Corollary 5.8 shows that if $\mathcal{W}$ is doubly invariant (i.e., $\mathcal{W}\mathcal{M} \subseteq \mathcal{W}$), then $\mathcal{W} = eL^\alpha(\mathcal{M},\tau)$ for some projection $e \in \mathcal{M}$.
• Corollary 5.9 establishes that if $\mathcal{W}$ is simply $H^\infty$-right invariant and $H^\infty \cap (H^\infty)^* = \mathbb{C}I$, then $\mathcal{W} = uH^\alpha$ for some unitary $u \in \mathcal{W} \cap \mathcal{M}$.