[Paper Review] Noncommutative multi-parameter Wiener-Wintner type ergodic theorem

Guixiang Hong, Mu Sun|arXiv (Cornell University)|Feb 2, 2016

Advanced Operator Algebra Research24 references3 citations

TL;DR

This paper establishes a noncommutative, multi-parameter extension of the Wiener-Wintner ergodic theorem for noncommutative trace-preserving dynamical systems. It introduces a weight class $\mathcal{D}$, strictly larger than the bounded Besicovitch class, and proves individual and uniform ergodic convergence for this class using noncommutative maximal inequalities and Orlicz space atomic decompositions.

ABSTRACT

In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener-Wintner type ergodic theorem for dynamical systems not necessarily being commutative. More precisely, we introduce a weight class $\mathcal{D}$, which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set $$Λ_d=\left\{\{λ^{k_1}_1\dotsmλ^{k_d}_d\}_{(k_1,\dots,k_d)\in \mathbb{N}^d}:\quad (λ_1,\dots,λ_d) \in \mathbb{T}^d ight\};$$ then prove a multi-parameter Bellow and Losert's Wiener-Wintner type ergodic theorem for the class $\mathcal{D}$ and for noncommutative trace preserving dynamical system $(\mathcal{M},τ,\mathbf{T})$. Restricted to consider the set $Λ_d$, we also prove a noncommutative multi-parameter analogue of Bourgain's uniform Wiener-Wintner ergodic theorem. The noncommutativity and the multi-parameter induce some difficulties in the proof. For instance, our arguments in proving the uniform convergence for a dense subset turn out to be quite different since the "pointwise" argument does not work in the noncommutative setting; to obtain the uniform convergence in the largest spaces, we show maximal inequality between the Orlicz spaces, which can not be deduced easily using classical extrapolation argument. Junge and Xu's noncommutative maximal inequalities with optimal order, together with the atomic decomposition of Orlicz spaces, play an essential role in overcoming the second difficulty.

Motivation & Objective

• To extend the classical Wiener-Wintner ergodic theorem to noncommutative, multi-parameter dynamical systems.
• To define and analyze a new weight class $\mathcal{D}$ that strictly contains the bounded Besicovitch class.
• To prove individual and uniform ergodic convergence for the class $\mathcal{D}$ in noncommutative $L_p$-spaces.
• To overcome challenges arising from noncommutativity and multi-parameter structure in ergodic convergence proofs.
• To establish a noncommutative analogue of Bourgain’s uniform Wiener-Wintner theorem for the set $\Lambda_d$.

Proposed method

• Introduce a weight class $\mathcal{D}$ using spectral measures and correlation functions, generalizing the bounded Besicovitch class.
• Utilize noncommutative maximal inequalities with optimal order from Junge and Xu to control suprema in Orlicz spaces.
• Apply atomic decomposition of Orlicz spaces to handle convergence in the largest possible function spaces.
• Establish uniform convergence on a dense subset via noncommutative maximal function estimates, bypassing pointwise arguments.
• Prove convergence of weighted ergodic averages $\frac{1}{n+1}\sum_{k=0}^{n} a(k) T^k(x)$ for $a \in \mathcal{D}$ in noncommutative $L_p$-spaces.
• Use spectral measure and Bohr-Fourier series techniques to characterize sequences in $\mathcal{D}$ via discrete spectral measures and absolutely convergent Fourier series.

Experimental results

Research questions

• RQ1 Can the Wiener-Wintner ergodic theorem be extended to noncommutative, multi-parameter dynamical systems?
• RQ2 What is the largest class of weights for which individual ergodic convergence holds in noncommutative $L_p$-spaces?
• RQ3 How can uniform convergence be established in noncommutative settings where pointwise arguments fail?
• RQ4 What role do noncommutative maximal inequalities and Orlicz space techniques play in extending the Wiener-Wintner theorem?
• RQ5 Is there a noncommutative analogue of Bourgain’s uniform Wiener-Wintner theorem for multi-parameter systems?

Key findings

• The weight class $\mathcal{D}$ is strictly larger than the bounded Besicovitch class, as demonstrated by a counterexample with $a(\mathbf{k}) = (-1)^{[\log(k_1+\cdots+k_d+1)]}$.
• For any $f \in L_p(\mathcal{M}, \tau)$ with $1 \leq p \leq \infty$, the weighted ergodic averages $\frac{1}{n+1}\sum_{k=0}^n a(k) T^k(f)$ converge in $L_p$-norm for all $a \in \mathcal{D}$.
• Uniform convergence holds for $a \in \mathcal{D}$ in the noncommutative $L_p$-setting, achieved via maximal inequalities in Orlicz spaces.
• The spectral measure $\sigma_\mathbf{a}$ of sequences in $\mathcal{D}$ is discrete and supported on $\mathbb{T}^d$, with $\sigma_\mathbf{a} = \sum_{\alpha=1}^\infty C_\alpha \delta_{\mathbf{z}_\alpha}$ and $\sum C_\alpha < \infty$.
• The amplitude $\Gamma_\mathbf{a}(\mathbf{z}) = \lim_n \frac{1}{|\mathbf{n}+1|} \sum_{\mathbf{k}=0}^{\mathbf{n}} a(\mathbf{k}) \bar{\mathbf{z}}^{\mathbf{k}}$ exists and satisfies $|\Gamma_\mathbf{a}(\mathbf{z})|^2 = \sigma_\mathbf{a}(\{\mathbf{z}\})$.
• The class $\mathcal{D}$ is characterized by conditions on the spectral measure, correlation function, and amplitude, with $\mathbf{a} \in \mathcal{D}$ iff $\gamma_\mathbf{a}(\mathbf{m}) = \sum_{\alpha=1}^\infty C_\alpha \mathbf{z}_\alpha^{\mathbf{m}}$ and $\sum C_\alpha < \infty$.