Dimension-Free Estimates for Noncommutative Discrete Maximal Spherical Means

TL;DR

The paper proves dimension-free $L_p$-estimates for operator-valued maximal spherical means on ${\mathbb Z}_{m+1}^d$ by extending the Nevo–Stein spectral method to the noncommutative setting. It introduces and analyzes vector-valued noncommutative $L_p$ spaces and noise-operator techniques to control both local and distant spherical averaging operators, ultimately establishing uniform bounds in the dimension $d$. As an application, the authors derive a noncommutative spherical maximal inequality for tau-preserving automorphism actions of cyclic groups on von Neumann algebras. The results unify transference principles with complex interpolation to yield dimension-free maximal inequalities for both operator-valued and fully noncommutative contexts, with potential impact on noncommutative ergodic theory and quantum harmonic analysis.

Abstract

In this paper, we establish dimension-free $L_p$-estimates for operator-valued maximal spherical means on cyclic groups $\Z_{m+1}^d$ for all $p>1$ and $m\geq1$. The key ingredient is a noncommutative extension of the spectral technique developed by Nevo and Stein. As an application, we obtain a noncommutative spherical maximal inequality for automorphism actions of von Neumann algebras.

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 2.1
• Remark 2.2
• Proposition 2.3
• Theorem 2.4: Lan76Y1JX
• Proposition 3.1
• Lemma 3.2
• proof